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5.1 DEFINITION OF CENTRE OF MASS Centre of mass: Every physical system has associated with it a certain point whose motion characterizes the motion of the whole system. When the system moves under some external forces than this point moves as if the entire mass of the system is concentrated at this point and also the external force is applied at his point of translational motion. This point is called the centre of mass of the system. Centre of mass of system of n point masses or n particles is that point about which moment of mass of the system is zero, it means that if about a particular origin the moment of mass of system of n point masses is zero, then that particular origin is the centre of mass of the system. Y The concept of centre of mass is a pure mathematical concept. If there are n  m (x , y ,z )   particles having mass m1, m2 ….m n and are placed in space (x1, y1, z1), (x2, y2, z2)    ……….(x n, y n, z n) then centre of mass of system is defined as (X, Y, Z) where  0(x, y,z)  O   X 1 i mi xi Y = 1 i mi yi and Z = 1 i mi zi where M = i mi is the total   =  M M M mass of X the system. Z Locate the point with coordinates (X, Y, Z). This point is called the centre of mass of the given collection of the particles. If the position vector of the i th particle is ri, the centre of mass is defined to have the position vector n   1 RCM =  mi ri Where M = m1 + m2 + ……….. + mn =  mi M i i1    r where r i  xi i  yi j  zi k 1

1

 i(m1x1  m2 x 2  ..  mx n )  j(m1y1  m2 y 2  ..  my n )  ..    i x cm  j y cm  k (z cm)  k(m1z1  m2 z2  ..  mzn )  n r mi ri  This is equation for centre of mass of n point masses. i=1 = M





1

1  m1(x1i+ y1 j + z1k) + m2 (x 2i+ y 2 j + z 2k) + ..   M  mn (xni+ yn j + znk) 



RCM =

RCM

1

1 M

Illustration 1: Solution:

Three equal masses are situates at vertices of equilateral triangle as shown in figure. Find centre of mass of the system. Let m be the mass of three masses and X CM, YCM and ZCM be the centre of masses of along the X axis, Y axis and Z axis, then

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XCM  XCM 

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m1x1  m2 x 2  m3 x 3 m1  m2  m3

y m

m  0   m  a   m  a / 2 mm m



a

a 2

3a / 2,0



a m

m (0,0,0)



 a / 2,

a



(a,0,0)

x

m1y1  m2 y 2  m3 y 3 m  0  m  0   m 3a / 2 a Y   CM m1  m2  m3 mm m 2 3 m z  m2 z 2  m3 z 3 m  0  m  0  m  0 ZCM  1 1 ZCM  0 m1  m2  m3 mm m YCM 

 a a  ,0 Hence coordinate of centre of mass is  ,  2 2 3  

5.2 CENTRE OF MASS OF CONTINUOUS BODIES If we consider the body to have continuous distribution of matter, the summation in the formula of centre of mass should be replaced by integration. If x, y, z are the coordinates of this small mass dm, we write the coordinates of the centre of mass as 1 1 1 X =  x dm, Y =  y dm, Z =  z dm M M M

y dm

 r x

O

The integration is to be performed under proper limits so that as the integration variable goes through the limits, the elements cover the entire body. We illustrate the method with three examples. Illustration 2:

Find centre of mass of a uniform straight rod of mass m and length l,

Solution:

Let M and L be the mass and the length of the rod respectively. Take the left end of the rod as the origin and the X–axis along the rod (figure). Consider an element of the rod between the positions A and B of the rod. Let the element be at a distance x from the centre O and its width (x, o, o) be dx. So as x varies from 0 through L, the elements cover x the entire rod. O

x

dx

As the rod is uniform, the mass per unit length is M/L and hence the mass of the element is dm = (M/L) dx. The x–coordinate of the centre of mass of the rod is 1 1  M  1  x 2 X =  x dm =  x  dx =   M M0  L  L  2 L

L

= 0

L 2

1 y dm =0 M  L  and similarly Z = 0. The centre of mass is at  ,0,0 , i.e. at the middle point of the rod.  2  The y–coordinate is ,

Y= Y=

5.3 MOTION OF CENTRE OF MASS Motion of the Center of Mass: Let us consider the motion of a system of n particles of individual masses m1, m2, ……., mn and total mass M. It is assumed that no mass enters or leaves the system during its motion, so that M remains constant. Then, as we have seen, we have the relation

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Chapter - 5 Rotational Motion

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r ur ur r ur ur r m1 r1  m2 r2  ......  mn rn m1 r1  m2 r2  ......  mn rn r CM   m1  m2  .......  mn M r r r Or Mr CM  m1 r1  m2 r 2  ......  mn Differentiating uu r this expression uu r uu r with respect uu r to time t, we have dr CM dr 1 dr 2 dr n M  m1  m2  ..... mn dt dt dt uu r dt dr Since,  velocity dt r r r r Therefore, Mv CM  m1 v1  m2 v 2  .....  mn v n … (i) Or velocity of the Center of Mass is n r r r r r m v m1 v1  m2 v 2  .....  mn v n  i i r Or v CM  i1 v CM  M M r ur Further, mv  momentum of a particle P . Therefore, Eq. (i) can be written as n u ur r ur ur ur ur PCM   Pi Or PCM  P1  P2  .....  Pn i 1

Differentiating respect we uur Eq. (i)uuwith r uur to time t, u ur get dv CM dv1 dv 2 dv n M  m1  m2  ......  mn dt dt dt dt r r r r Or MaCM  m1 a1  m2 a2  ......  mn an r r r r m1 a1  m2 a2  .....  mn an Or aCM  M n r m1 a1  r Or i 1 aCM  M Further, in accordance with Newton’s second law of motion n r r r r r FCM   Or FCM  F1  F2  .....  Fn

… (ii)

r r F  m a . Hence, Eq. (ii) can be written as r Fi

i 1

Thus, as pointed out earlier also, the centre of mass of a system of particles moves as though it were a particle of mass equal to that of the whole system with all the external forces acting directly on it. Illustration 3:

Solution:

Two particles A and B of mass 1 kg and 2 kg respectively are projected in the directions shown in figure with speeds uA  200 m/s and uB  50 m/s. Initially they wee 90 m apart. Find the maximum height attained by the centre of mass of the particles. Assume acceleration due to gravity to be constant. (g  10 m/s2). Using mArA  mBrB B Or (1)(rA)  (2)(rB) Or rA  2rB … (i) uB 90m And rA  rB  90 m … (ii) uA Solving these two equations, we get rA  60m and rB  30m A i.e., CM is at height 60m from the ground at time t  0. r r r m A a A  mB aB aCM  Further,  g  10 m/s2 (downwards) m A  mB r r As (downwards) a A  aB  g

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r r r m A uA  mB uB  1  200   2  50  100 uCM    m / s (upwards) m A  mB 1 2 3 Let, h be the height attained by CM beyond 60 m. Using 2 2 v CM  uCM  2aCMh 2

 100   2  10 h  3

Or 0   

Or

2 100   h  55.55m

180 Therefore, maximum height attained by the centre of mass is , H  60  55.55  115.55 m

5.4 CONSERVATION OF LINEAR MOMENTUM ur The product of mass and the velocity of a particle is defined as its linear momentum P . So,

 

ur r Pmv

The magnitude of linear momentum may be written as P  mv  1  P2  m2 v 2  2m  mv 2  2mK Or  2 

P2 2m Here, K is the kinetic energy of the particle. In accordance with Newton’s second law, r uur uur r r dv d mv dv F  ma  m   dt dt dt uuu r r dP Thus, F dt In case the external uuu rforce applied to a particle (or a body) be zero, we have r dP ur F 0 or P  cons tan t dt Showing that in the absence of an external force, the linear momentum of a particle (or the body) remains constant. This is called the law of conservation of linear momentum. The law may be extended to a system of particles or to the centre of mass of a system of particles. For example, for a system of particles it takes the form. If net force (or the vector sum of all the forces) on a system of particles is zero, the vector sum of linear momentum of all r the r particles r r remain r conserved, or If F  F1  F2  F3  .....  Fn  0 ur ur ur ur Then, P1  P2  P3  .....  Pn  constant The same is ther case for ur the centre of mass of a system of particles i.e., if FCM  0,PCM  Constant Thus, the law of conservation of linear momentum can be applied to a single particle, to a system of particles or even to the centre of mass of the particles. Thus,

P  2Km or K 

 

Illustration 4: Solution:

A man of mass m1 is standing on a platform of mass m2 kept on a smooth horizontal surface. The man starts moving on the platform with a velocity v r relative to the platform. Find the recoil velocity of platform. Absolute velocity of man  r   where   recoil velocity of platform. Taking the platform and the man as a system, net external force on the system in horizontal direction is zero. The linear momentum of the vr - v system remains constant. Initially both the man and the v platform were at rest.

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Hence, from law of conservation of momentum 0  m1 (vr – v) – m2v m1v r  v m1  m2

IMPULSE r r Consider a constant force F which acts for a time t on a body of mass m, thus, changing its velocity from u r r to v . Because the force is constant, the body will travel with travel with constant acceleration a where r r r r r and at  v  u F  ma r Ft r r Hence,  vu m r r r Or Ft  mv  mu r r The product of constant force F and the time t for which it acts is called the impulse J of the force and this is equal to the change in linear momentum which it produces. r r r r r Impulse J  Ft  P  Pf  Pi





Instantaneous Impulse: There are many occasions when a force acts for such a short time that the effect is instantaneous, e.g., a bat striking a ball. In such cases, although the magnitude of the force and the time for which it acts may each be unknown but the value of their product (i.e., impulse) can be known by measuring the initial and final momenta. Thus, we can write r r r r r J   Fdt  P  Pf  Pi Regarding the impulse it is important to note that impulse applied to an object in a given time interval can also be calculated from the area under force–time (F – t) graph in the same time interval.

Impulse momentum relation: The change in momentum of a particle during a time interval equals the impulse of the net force that acts on the particle during that interval. r r r (impulse – momentum relation) J  p2  p1 Illustration 5: Solution:

A ball of mass m, travelling with velocity 2iˆ  3ˆj receives an impulse – 3m ˆi . What is the velocity of the afterwards? r ballrimmediately r Using J  m(v f  v i ) r r r v f  3iˆ  2iˆ  3ˆj –3m ˆi = m  v f  2iˆ  3ˆj Or Or v f   ˆi  3ˆj









5.5 VARIABLE MASS In our discussion of the conservation of linear momentum, we have so far dealt with system whose mass remains constant. We now consider those systems whose mass is variable, i.e., those in which mass enters or leaves the system. A typical case is that of the rocket from which hot gases keep on escaping, thereby continuously decreasing its mass. r In such problems you have nothing to do but apply a thrust force Ft to the main mass in addition to the all r r  dm Ft  v rel  other forces acting on it. This thrust force is given by,  dt  r  Here, v rel is the velocity of the mass gained or mass ejected relative  v    v v  dv to the main mass. In case of rocket this is sometimes called the dm dm m - dm exhaust velocity of the gases. is the rate at which mass is system dt

 

r

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increasing or decreasing. The expression for the thrust force can be derived from the conservation of linear momentum in the absence of any external forces on a system as follows : r Suppose at some moment t  t mass of a body is m and its velocity is v . After some time at t  t  dt its mass r uur r becomes (m  dm) and velocity becomes v  dv . The mass dm is ejected with relative velocity v r , Absolute r r uur velocity of mass ‘dm’ is therefore v r  v  dv . If no external forces are acting on the system, the linear





momentum of the system will remain conserved, r r uur r r uur ur ur m v   m  dm v  dv  dm v r  v  dv Or Or Pi  Pf ur r r r uur r r uur m v  m v  md v  dm v   dm dv  dm v  v r dm   dm  dv Or r r  mdv   v r dm uur r  dm  dv m    vr    Or  dt  dt uur r  dv m    thrust force F1 Here,  dt dm  And = rate at which mass is ejecting dt

  







 

 

Problems related to variable mass can be solved in following three steps: 1. Make a list of all the forces acting on the main mass and apply them on it. r  dm r 2. Apply an additional thrust force Ft on the mass, the magnitude of which is v r    and direction  dt r r is given by the direction of v r in case the mass is increasing and otherwise the direction of  v r if it is decreasing. 3. Find netuforce on the mass and apply ur r dv (m = mass at that particular instant) Fnet  m dt Rocket Propulsion: Let m0 be the mass of the rocket at time t = 0. m its mass at any time t and v its velocity a that the velocity of the rocket is u.

At

At

t=0 v=u m = m0

t=t m=m v=v

Exhaust velocity = vr   dm be the mass of the gas ejected per unit time and vr the exhaust velocity of the gases. Usually  dt 

Further, let  

  dm and vr are kept constant throughout the journey of the rocket. Now, let us write few equations which can  dt   be used in the problems of rocket propulsion. At time t = t, 1.

Thrust force on the rocket

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  dm  dt   Weight of the rocket W = mg New force on the rocket Fnet = Ft – W   dm Fnet  v r   mg Or  dt   Ft  v r 

2. 3.

4.

(upwards) (downwards) (upwards)

Net acceleration of the rocket a  Or

dv v r  dm  g  dt m  dt  v

Or

 dv  v r u

Thus, Illustration 6:

Solution:

m



m0

AIEEE

Or

F m  dm  gdt  m 

dv  v r 

t

 dm  g  dt m 0

Or

 m v  u  gt  v r In  0  m

 m v  u  v r In  0  gt  m … (i)

(A) A rocket set for vertical firing weighs 50 kg and contains 450 kg of fuel. It can have a maximum exhaust velocity of 2 km/s. What should be its minimum rate of fuel consumption (i) to just lift it off the launching pad. (ii) to give it an acceleration of 20 m/s2. (B) What will be the speed of the rocket when the rate of consumption of fuel is 10 kg/s after whole of the fuel is consumed ? Take g = 9.8 m/s2. (A) To just lift it off the launching pad Weight = thrust force mg   dm   dm  mg  v r  Or Or  dt    dt   vr

 450  50  9.8  2.45kg / s   dm   dt   2  103 (ii) Net acceleration a = 20 m/s2 v   dm F a r  g ma = Ft – mg Or a t g Or  m  dt  m m  g  a   dm  This gives    dt  vr Substituting the values, we get,

 450  50  9.8  20  7.45kg / s   dm   dt   2  103 (B) The rate of fuel consumption is 1.0 kg/s. So, the time for the consumption of 450  45 second entire fuel is, t  10  m0 Using Eq. (i), i.e., v  u  gt  v r In   m  3 Here, u = 0, vr = 2  10 m/s, m0 = 500 kg and m = 50 kg Substituting the values, we get,  500 v  0  (9.8)(45)  (2  103 )In  Or v = – 441 + 4605.17  50  Substituting the values, we get,

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Or

v = 4164.17 m/s

Or

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v = 4.164 km/s

5.6 COLLISIONS When two particles approach each other, their motion changes or their momentum changes due to their mutual interactions. This phenomenon is called collision. During collision (i) an impulse (a large force for a relatively short time) acts on each colliding particle (ii) the total momentum of the particles remain conserved. The collision is infact a redistribution of total momentum of the particles. Physical interaction is not necessary for collision. Generally, the collisions are of two types: (1) Elastic collision (2) Inelastic collision 1. Elastic collision: A collision is said to be elastic if kinetic energy is also conserved along with the linear momentum. There is no loss or transformation of kinetic energy in this collision uu r uu r Consider two particles of masses m1 and m2 moving with velocities u1 and u2 respectively. They collide uu r uur and their velocities after the collision become v1 and v 2 respectively. By conservation of the linear momentum, Total linear momentum = Total linear momentum after uu r uu r before uu r collision uur collision , m1u1 + m2 u2 = m1 v1 + m2 v 2 uu r uu r uur uu r m1 u1 - v1 = m2 v 2 - u2 or … (1)









By conservation of kinetic energy, Total kinetic energy before collision = Total kinetic energy after collision 1 1 1 1 m1u12 + m2u22 = m1v12 + m2 v 22 2 2 2 2 2 2 2 2 m1 u1 - v 1 = m2 v 2 - u2 or … (2)





u1'





and only for one dimension collision or head on collision uu r uu r uu r uur On solving the above equation (1) and (2), u1 - u2 = - v 1 - v 2





u'2 u1

u2

Long before collision

 

v v1 v

' 1

 During  collision 

v2

 v

' 2



Long after collision

… (3)

i. e. Relative velocity before collision = Relative velocity after collision. Thus in elastic collision, the relative velocity of approach of particles before collision is equal to the relative velocity with which the particles recedes after collision. i.e. the magnitude of relative velocity remains the same, but the direction is reversed. On solving the equation (1) and (3) we get  m - m2   2m1   m2 - m  1 2m2 v1 =  1 u1 + u2 and v 2 =  u1 + u2     m1 + m2  m1 + m  2  m1 + m2  m1 + m  2 Illustration 7: Solution:

A ball of 0.4kg mass and a speed of 3 m/s has a head-on, completely elastic collision with a 0.6-kg mass initially at rest. Find the speeds of both bodies after the collision: By Conservation of momentum: (0.4  3) + 0 = 0.4v + 0.6V or v + 1.5V = 3 We know that: velocity of separation = –velocity of approach or –v + V = 3 We solve by adding the two equations to yield 2.5V = 6 V = 2.4 m/s v = –0.6 m/s

2. Inelastic collision: A collision in which the linear momentum is conserved, but a part of kinetic energy change into the other forms (such as heat, vibration, excitation energy etc) is called the inelastic collision. In other words, the kinetic energy is not conserved.(Actually there is no violation of the law of conservation of energy, but a part of the kinetic energy, changes into a useless form.) In this, the particles do not regain their shape and size completely after collision. Some fraction of mechanical energy is retained by the

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colliding particles in the form of deformation potential energy. However, in the absence of external forces, law of conservation of linear momentum still holds good. uu r uu r Let two particles of masses m1 and m2 moving with initial velocities u1 and u2 uu r uur Collide and travel with velocities v1 and v 2 respectively after the collision. By conservation of linear momentum uu r uu r uu r uur m1u1 + m2 u2 = m1 v1 + m2 v 2  uu r uu r uur uu r  or m1 u1 - v1 = m2 v 2 - u2  1 1 1 1 and by conservation of kinetic energy, m1u12  m2u22 = m1v12  m2 v 22 + E 2 2 2 2 Where E is the part of energy which changes into the useless form due to inelastic collision; r For example, if two particles coalesce after collision and the combined system travel with a velocity v after the collision in same direction, then m u  m 2 u2 m1u1  m2u2   m1  m2  v or v  1 1  m1  m2 





And loss in kinetic energy, E =





1 1 1 m1u12  m2u22   m1  m2  v 2 2 2 2

1 1  m u  m2u2 E= m1u12  m2u22   1 1 2 2   m1  m2   



or



2



1 m1m2 (u  u2 )2 2 (m1  m2 ) 1

5.7 CONCEPT OF COEFFICIENT OF RESTITUTION When two bodies collide head–on, the ratio of their relative velocities after collision and their relative velocities before collision is called the coefficient of restitution e. r r v 2 - v1 e= r r Thus … (i) u1 - u2

seperation speed (along line of impact) … (ii) approach speed Where v1 = The speed of first body after collision. v2 = The speed of second body after collision. u1 = the speed of first body before collision. u2 = the speed of second body before collision. The ratio e is called the coefficient of restitution and is constant for two particular objects. 0  e 1 In general, e = 0, for completely inelastic collision, as both the objects stick together. So, their separation speed is zero or e = 0. e = 1, for an elastic collision, as we can show that from equation (i) v2 – v1 = u1 – u2 … (iii) Or separation speed = approach speed Or e=1 Let us now find the velocities of two particles after collision if they collide directly and the coefficient of restitution between them is given as e. e

Or

m2

m1

v2

Before Collision

v1

v2'

v1'

After Collision

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Applying conservation of linear momentum, m1v1  m2 v 2  m1v1'  m2 v 2' Further, separation speed = e (approach speed) or u1 – u2 = e (v2 – v1)  m  em2  m1  em  2 ' v1  v2 Solving equations (iv) and (v) we get , v1   1  m1  m 2  m1  m2   and

… (iv) … (v) … (vi)

 m2  em1  m1  em  1 v 2  v1  m1  m 2  m1  m2  

v 2'  

Illustration 8: Solution:

… (vii)

A moving particle of mass m, makes a head–on collision with a particle of mass 2m, which is initially at rest. Show that the colliding particle loses (8/9) of its energy after collision. Let u be the initial velocity of particle of mass m and  its velocity after the collision. Let V be the velocity of particle of mass 2m after the collision. From the principle of conservation of linear momentum, we have mu = mv + (2m) V Or u–v=2V … (i) The conservation of kinetic energy gives 1 1 1 mu2 = mv2 + (2m)V2 Or u2 – v2 = 2V2 … (ii) 2 2 2 Or (u – v) (u + v) = 2V2 Using Eq. (1) in Eq. (2) we have 2V (u + v) = 2V2 or u + v = V Or 2(u + v) = 2V … (iii) u Comparing (1) and (3) we get, u – v = 2 (u + v) Or v =  … (iv) 3 1 Now, initial kinetic energy of the colliding mass is, Ki = mu2 2 1 Final kinetic energy, Kf = mv2 2 1 1 Loss in kinetic energy is, K = Ki – Kf = mu2 – mv2 2 2 1 1 2 mu2 - mv 2 K u2 - v 2  v 2  Fractional loss = = 2 = = 1 –  u 1 Ki  u2 mu2 2 2 8  1 =1–    = (Q v = – u/3)  3 9

5.8 OBLIQUE COLLISION When two bodies collide such that there velocities are not in the line of action of contact force then the collision is known as oblique collision. We can divide also oblique collision in elastic, and inelastic collision. It is important to notice that momentum will conserve in each direction curing oblique collision also. When two bodies collide obliquely, their relative velocity resolved along their common normal after impact is in a constant ratio to their relative velocity before impact (resolved along common normal) and is in the opposite direction. v1 cos   v 2 cos   e  v1 cos   v 2 cos    e  u1 cos   u2 cos   u1 cos   u2 cos   u1

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 u2

  v1

v2

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Illustration 9:

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A ball of mass m moving at a speed v makes a head on collision with an identical ball at rest. The energy of the balls after the collision is 3/4th of the original. Find the coefficient of restitution. m

v2'

m

v

v1'

After Collision

Before Collision

Solution:

As we have seen in the above discussion, that under the given conditions :  1 e  1 e v1'   v and v 2'  v   2  2  3 Given that K f  K i 4

Or

2 1 3 1  mv1'   mv 2  2 4 2

2

Substituting the value, we get,

 1 e  1 e   2     2 

Or (1 + e)2 + (1 – e)2 = 3 Or

2 + 2e2 = 3

Or

e

2

Or



3 4 e2 

1 2

1 2

5.9 RIGID BODY A rigid body is a body with a definite and unchanged shape and size i.e. a body is said to be rigid if the distance between any two particles of the body remains invariant.

MOTION OF A RIGID BODY Translational: If a body moves such that its orientation does not change with respect to time then body is said to move in translational motion.

ROTATION: If a body is rotating about the fix axis of rotation then its motion is known as pure rotational motion. MIX – MOTION: If a body moves such that its motion neither be pure rotational nor be pure translational then its motion is known as mix motion.

5.10 ANGULAR DISPLACEMENT Consider a rigid body undergoing rotation about an axis, perpendicular to the plane of the paper and passing through O. Suppose that A and B are any two particles of the rigid body at the position 1 while A and B are their subsequent locations when the body is at the position 2. Since the body undergoes rotation, OA = OA and OB = OB Further AB = AB, since the body is rigid.  OAB  OAB(congruent) i.e.  AOB = AOB Adding  AOB to both sides of the above equation, we get

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 BOB = AOA =  (say) This implies that in a given interval of time the angular displacements of all particles of the rigid body undergoing rotation are identical. Therefore, a single variable, viz. angular displacement () can be used to describe the rotational motion of the rigid body. Angular displacement is not a vector quantity

5.11 ANGULAR VELOCITY( ) The rate of change of angular displacement with respect to time is known as angular velocity. Average angular velocity: The rate of change of angular displacement with respect to time is known as angular velocity. To define average angular velocity it is necessary to specify the interval in which we are talking about 2  1 average.  = t 2  t1 Instantaneous angular velocity: Instantaneous velocity means angular velocity at a particular instant. It is   1 d    lim 2  mathematically defined as ,  at t  t1 t 2  t1 t  t dt  2 1

5.12 ANGULAR ACCELERATIONS ( ) Angular accelerations (): The rate of change of angular velocity with respect to time is known as angular acceleration. Average angular acceleration: It is necessary to specify the time interval to in which we are talking about average. r r r  2  1   In a given time interval t1 to t2 the average angular acceleration is defined as , t 2  t1 Instantaneous angular acceleration: Instantaneous angular acceleration means angular acceleration at a r r r  2  1 d     limit  particular dot instant at t = t1 mathematically it is defined as,  t1  t 2 t 2  t1 dt  t  t 2

Direction of angular acceleration: If magnitude of  increasing then direction of  will be same as direction of  and vice–versa.

5.13 EQUATION OF ANGULAR MOTION (t) = o + t (t) = o + ot +

1 2 t 2

2

2   t  =  0  2

Here

o = magnitude of the initial angular velocity (t) = magnitude of the angular velocity after time t. o = Initial angular position. (t) = Angular position after time t.

Illustration 10:

A disc starts rotating with constant angular acceleration of  radian/s2 about a fixed axis perpendicular to its plane and passing through its centre. Find (a) The angular velocity of the disc after 4 sec. (b) The angular displacement of the disc after 4 sec and

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Solution:

(c) Here (a) (b) (c)

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Number of turns accomplished by the disc in 4 sec. =  rad/sec2 0 = 0 t = 4 sec (4 sec) = 0 + ( rad/sec2)  4 sec = 4 rad/sec. 1 (4 sec) = 0 + ( rad/sec2)  (16 sec2) = 8  radian. 2 Let the number of turns be n  n 2  rad = 8  rad  n=4

5.14 RELATION BETWEEN LINEAR AND ANGULAR VARIABLES Consider a particle A of a rigid body undergoing rotation about a fixed axis- nˆ , the particle A describing an arc ABA of a circle with its centre O on the axis of rotation. Taking the origin at O, the position uuurvector of A, r r  OA . OA = OA = constant (radius of the circle) AOA =  (t) (say) The arc length, ABA, S = r dS d  r = r The tangential velocity, vA = dt dt r The direction of the angular velocity vector  be taken along the axis of rotation: r  =  nˆ , nˆ being the unit vector along the axis of rotation. r Then, v , instantaneous velocity of A with respect to the axis of rotation, can be written as , The acceleration of the point A with respect to the axis of rotation is r r r r dv d r r dr r r r r r  r    a= ar v dt dt dt r r r r r r r r r If  is constant, then  = 0 and a    v =      r  =  2 r

r r r vr

5.15 TORQUE Torque of a force about the axis of rotation: The turning effect of a force P F about the axis of rotation is called moment of force or torque due to the force and is measured as the product of the magnitude of the force and the r perpendicular distance of the line of action of the force from the axis of Fig. (a) O rotation. It is denoted by the Greek letter  (tou). Thus, torque = force  perpendicular distance from the axis of rotation Its unit is N m in SI. Its dimensional formula is [M L 2 T–2], same as that of work. It may be pointed out that no doubt, N m is equivalent to joule (the unit of work); but joule is not used as the r r unit of torque. Consider a force F acting on a particle P. Choose an origin O and let r be the position vector of the ur r r particle experiencing the force. We define the torque of the force F about O as , P  r  F … (1) r r A This is a vector quantity having its direction perpendicular to r and F according to the Fig. (b) rule of cross product. Now consider a rigid body rotating about a given axis of rotation AB (fig b). Let F be a force acting on the particle O of the body. F may not be in the F P O plane ABP. Take the origin O somewhere on the axis of rotation. r O 1

B

ur r r r The torque of F about O is P  r  F . Its component along OA is called the torque of F about OA. To r r calculate it, we should find the vector r  F and then find out the angle  it makes with OA. The torque S tratf ord Acad emy is an Id eal In sti tu te For IIT-JEE & AIE EE Coach i n g Con tact : 0731-6541611, 98267- 52000

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r r about OA is then r  F cos  . The torque of a force about a line. This can be shown as given below. Let O 1 be any point onrthe line r 10.4). r torque r ABr (Figure r The r r of Frabout O1 is O1P  F  (O1O  OP)  F  O1O  F  OP  F . r r r As O1P  F  O1O, this term will have no component along AB. r r r r Thus, the component of O1P  F is equal to that of O1P  F . Illustration 11: Solution:

A particle of mass m is dropped at point A, find the torque about O. r r r   r F r    r F sin nˆ = (r sin) F nˆ   = b mg The direction of torque is directed inward the paper or in other words, rotation about O is clockwise.

O

A

b

m r

 mg

5.16 KINETIC ENERGY OF A RIGID BODY Let a rigid body is purely rotating about an axis AB with angular velocity  consider a general particle m2 which is at a distance of r2 from axis of rotation. V2 = r2. So energy associated with this m2 is K · E2  1 1 A K · E2 = m2 v 22  m2 r22 2 2 2 r2 1 1 m1 K·Etotal =  k·E2   m2 ·r22 2   2  m2r22 2 2 1 2 I K·Etotal = 2 Where I is called as moment of inertia of body about an given axis of rotation. In this case I is about AB. Moment of inertia is also called as rotational mass of object. Illustration 12:

Solution:

Four point masses each of value m are placed at points A, B, C and D at distances a 3a ,a, and 2a from the free and of a mass-less rod. (i) Find the M.I. of the system about 2 2 an axis perpendicular to the point B. (ii) Will the mass on the left side contribute a negative term? Suppose that the four masses each of mass m are placed at the points A, B, C and D of a mass-less rod (fig.) (i) M.I. of the rod about the axis through the point B, I = m (BA)2 + m (BB)2 + m (BC)2 + m (BD)2 = m (a/2)2 + m (0)2 + m (a/2)2 + m (a)2 1 1 3 = m a2  0  m a2  m a2  m a2 2 4 4 (ii) The distance occurs with power 2 in the formula for moment of inertia. AXIS Therefore, if the distance of the mass on left of the axis is taken as negative, its moment of inertia will still m m m m contribute a positive term. Hence, the mass m on the A B C D left side of the axis will not contribute a negative term a/2 to the moment of inertia of the system. 3a/2 2a

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5.17 MOMENT OF INERTIA Like the centre of mass, the moment of inertia is a property of an object that is related to its mass distribution. The moment of inertia (denoted by I) is an important quantity in the study of system of particles that are rotating. The role of the moment of inertia in the study of rotational motion is analogous to that of mass in the study of linear motion. Moment of inertia gives a measurement of the resistance of a body to a change in its rotational motion. If a body is at rest, the larger the moment of inertia of a body, the more difficult it is to put that body into rotational motion. Similarly, the larger the moment of inertia of a body, the more difficult it is to stop its rotational motion. Moment of Inertia of a Single Particle: For a very simple case the moment of inertia of a single particle about an axis is given by, I = mr2 Here, m is the mass of the particle and r its distance from the axis under consideration.

r

Moment of Inertia of a System of Particles The moment of inertia of a system of particles about an axis is given by : I   miri2

m

r1

m1

r2

i

m2

r3

m3

Where ri is the perpendicular distance from the axis to the ith particle, which has a mass mi. Moment of Inertia of Rigid Bodies: For a continuous mass distribution such as found in a rigid body, we replace the summation of Eq. (ii) by an integral. If the system is divided into infinitesimal elements of mass dm and if r is the distance from a mass element to the axis of rotation, the moment of inertia is, I   r 2 dm

r dm

Where the integral is taken over the system.

Radius of Gyration: Radius of gyration may be defined as the distance from the axis at which, if the whole mass of the body were to be concentrated, the moment of inertia would be the same about the given axis as with its actual distribution of mass. Suppose a rigid body consists of n particles of each of the mass m. Let r 1, r2, ...... rn be the perpendicular distances of these particles from the axis of rotation. Then I  mr12  mr22  .....  mrn2  m  r12  r22  .....  rn2  r12  r22  .....  rn2  n  

M

 r12  r22  .....  rn2  n  

mn

(where M = m  n)

… (i)

If whole mass of the body is regarded to be concentrated at a perpendicular distance K (radius of gyration), then I = M K2 … (ii) From eqs. (1) and (2),

 r 2  r 2  .....  rn2 K 1 2  n  

1 2

… (iii)

Therefore, radius of gyration of a body about an axis is equal to the root mean square distance of the constituent particles from the given axis.

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THEOREM OF PERPENDICULAR AXIS Illustration 13:

Solution:

The moment of inertia of a thin square plate ABCD (as shown in figure) 4 of uniform thickness about an axis passing through the centre O and 1 A B perpendicular to the Plane of plate is: (A) I1 + I2 (B) I3 + I4 O 3 (C) I1 + I3 (D) I1 + I2 + I3 + I4 Where I1, I2, I3 and I4 are moments of inertia about axis 1, 2, 3 D C 2 and 4 respectively which are in the plane of plate? If I0 is moment of inertia of plane passing through the centre and  to plate, then according to the theorem of perpendicular axes. I0 = I1 + I2 = I3 + I4 From symmetry, I1 = I2 and I3 = I4  I0 = 2I2 = 2I3 i.e., I2 = I3  I0 = I1 + I2, I0 = I3 + I4 And I0 = I1 + I3 i.e., first three answers are correct. Ans. (A, B, C)

THEOREM OF PARALLEL AXES The moment of inertia of a body about any axis is equal to its moment of inertia about a parallel axis through its centre of gravity plus the product of the mass of the body and the square of the perpendicular distance between the two parallel axes. Illustration 14:

Solution:

The moment of inertia of a ring about one of its diameters is I. What will be its moment of inertia about a tangent parallel to the diameter? 3 I (A) 4I (B) 2I (C) (D) 3I 2 1 I  MR2 . According to theorem of parallel axes. (D) 2 1 3 I'  MR2  MR 2  MR 2  3I. 2 2

5.18 LAW OF ROTATION If a body rotates purely about an axis S with angular acceleration  and net torque acting on the body about s is s, then – s = I 

S

Illustration 15:

Solution:

A triangular plate of uniform thickness and density is made to rotate about an axis perpendicular to the plane of the paper and (a) passing through A, (b) passing through B, by the application of the same force, B C F, at C (midpoint of AB) as shown in the figure. Is the angular A acceleration in both the cases be the same ? F    = I  I  = Force × perpendicular distance Torque is same in both the cases. But since I will be different due to different mass distribution about the axis.   will be different for different cases

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5.19 ROTATIONAL WORK AND ENERGY The rotational work done by a force about the fixed axis of rotation is defined as , Wrot =  d Where  the torque is produced by the force, and d is the infinitesimally small angular displacement about the axis. 1 The rotational kinetic energy of a body about a fixed rotational axis is defined as, K rot  I 2 2 Where I is the moment of inertia about the axis. Work–Energy Theorem: In complete analog to the work energy theorem for the translator motion, it can be stated for rotational motion as: Wrot =  Krot The net rotational work done by the forces is equal to the change in rotational kinetic energy of the body. Conservation of Mechanical Energy: In the absence of dissipative work done by non–conservative forces, the total mechanical energy of a system is conserved. K  U  0 Or Kf + Uf = Ki + Ui

5.20 ANGULAR MOMENTUM OF A PARTICLE ABOUT A POINT AND ABOUT AN AXIS Angular momentum: The turning movement of a particle about the axis of rotation is called the angular momentum of the particle and is measured as the product of the linear momentum and the perpendicular distance of its lien of action from the axis of rotation. It is denoted by L. Therefore, Angular momentum = linear momentum  perpendicular distance from the axis of rotation In SI, the unit of angular momentum is kg m2 s–1. Its dimensional formula is [M L2 T–1]

ANGULAR MOMENTUM OF A PARTICLE ABOUT A POINT ur r r Suppose a particle A of mass m is moving with linear momentum P  m v . Its angular momentum L about r r ur r r r r L  r  P  r  mv  m r  v point 0 is defined as: r A  Here, r is the radius vector of particle A about 0 at that instant of time. The r   magnitude of L is P  mv  r  r sin  L = mvr sin  = mvr r Here, r= r sin  is the perpendicular distance of line of action of r r r r Velocity v from point O. The direction of L is same as that of r  v . 

 





O

Illustration 16:

Solution:

A particle of mass m is projected from origin O with speed u at an angle  with positive x-axis is in vertically upward direction. Find the angular momentum of particle at any time t about O before the particle strikes the ground again. r r r P L m r  v  y v r  r  t   xiˆ  yjˆ Here,



and 



1     u cos   tiˆ   ut sin   gt 2 ˆj  2  r v  t   v x ˆi  v y ˆj = (u cos ) ˆi + (u sin  – gt) ˆj r r r L m r  v



r

r  b

 O

x



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ˆi  m  u cos   t u cos 



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ˆj

kˆ 1 2  u sin  t  gt 0 = m[(u2 sin  cos ) t – (u cos ) gt2 2 0 u sin   gt



 u2 sin  cos  t 

1  u cos   gt 2  kˆ   1 m  ucos  gt 2 kˆ 2 2 

ANGULAR MOMENTUM OF A RIGID BODY ABOUT AN AXIS Suppose a particle P of mass m is going in a circle of radius r and at some instant the speed of the particle is v. For finding the angular momentum of the particle about the axis of rotation, the origin may be chosen anywhere on the axis. We choose itraturthe r ur centre of the circle. In this case r and P are perpendicular to each other and r  P is r ur along the axis. Thus, component of r  P along the axis is mvr itself. The angular momentum of the whole rigid body about AB is sum of components of all particles, i.e. L   miri v i

A



r O

P

B

i

Here, 

vi = ri L   miri2 i

or

L    miri2 i

or

L = I

(as

mr

i i

i

2

I )

RELATION BETWEEN TORQUE AND ANGULAR MOMENTUM

r r r L  r p r r r r r r r r r dL dr r r dp r   p r   v   mv   r  F = 0 + r  F   ext dt dt dt r r This relation is analogous to F  ma , which is applied in rotation. 



r dL r   ext dt

CONSERVATION OF ANGULAR MOMENTUM uur r dL We know that   dt

r r dL When there is no net external torque acting on a particle, then  0.  L = constant dt Therefore, the angular momentum of the particle remains invariant in the absence of any net external torque.

ANGULAR IMPULSE t2

The angular impulse of a torque in a given time interval is defined as

r

  dt t1

uur r dL r Hence,  is the resultant torque acting on the body. Further, since   dt t2 r r r Or   dt  angular impulse  L 2  L1

r uur   dt  dL

t1

Thus, the angular impulse of the resultant torque is equal to the change in angular momentum. Let us take an example based on the angular impulse.

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Illustration 17:

Solution:

Chapter - 5 Rotational Motion

AIEEE

The torque  on a body about a given point is found to be equal to A × L where A is constant vector, and L is the angular momentum of the body about that point. From this it follows that dL (A) is perpendicular to L at all instants of time dt (B) the component of L in the direction of A does not change with time. (C) the magnitude of L does not change with time. (D) L does not r change with time r dL (A, B, C)   dt r dL r r r r r Given that,   A  L   A L dtuur r r dL From cross–product rule, is always perpendicular to the plane containing A and L dt r r By the dot product definition L.L  L2 r r r dL r dL dL Differentiating with respect to time L.  L.  2L dt dt dt r r dL dL  2L.  2L dt dt r r dL dL Since is perpendicular to L  = 0  L = constant dt dt

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ASSIGNMENT LEVEL - I 1.

The coefficient of restitution e for a perfectly elastic collision is  (A) 1 (B) 0 (C) (D) –1 A shell initially at rest explodes into two pieces of equal mass, then the two pieces will (A) Be at rest (B) Move with different velocities in different directions (C) Move with the same velocity in opposite directions (D) Move with the same velocity in same direction Two perfectly elastic particles P and Q of equal mass travelling along the line joining them with velocities 15 m/sec and 10 m/sec. After collision, their velocities respectively (in m/sec) will be (A) 0, 25 (B) 5, 20 (C) r 10, 15 (D) 20, 5 A particle of mass m moving with a velocity V makes a head on elastic collision with another particle of same mass initially at rest. The velocity of the first particle after the collision will be r r r (A) (B) (C) (D) Zero V V 2V If two balls each of mass 0.06 kg moving in opposite directions with speed 4 m/s collide and rebound with the same speed, then the impulse imparted to each ball due to other is (A) 0.48 kg-m/s (B) 0.24 kg-m/s (C) 0.81 kg-m/s (D) Zero An inelastic ball is dropped from a height of 100 m. Due to earth, 20% of its energy is lost. To what height the ball will rise (A) 80 m (B) 40 m (C) 60 m (D) 20 m

2.

3. 4.

5. 6.

7. 8. 9. 10. 11.

12.

13. 14.

Which of the following is not a perfectly inelastic collision (A) Striking of two glass balls (B) A bullet striking a bag of sand (C) An electron captured by a proton (D) A man jumping onto a moving cart If a force of 250 N act on body, the momentum acquired is 125 kg-m/s. What is the period for which force acts on the body (A) 0.5 sec (B) 0.2 sec (C) 0.4 sec (D) 0.25 sec A body, whose momentum is constant, must have constant (A) Force (B) Velocity (C) Acceleration (D) All of these A rocket of mass 1000 kg exhausts gases at a rate of 4 kg/sec with a velocity 3000 m/s. The thrust developed on the rocket is (A) 12000 N (B) 120 N (C) 800 N (D) 200 N A man fires a bullet of mass 200 g at a speed of 5 m/s. The gun is of one kg mass, by what velocity the gun rebounds backwards (A) 0.1 m/s (B) 10 m/s (C) 1 m/s (D) 0.01 m/s Two objects of masses 200 gm and 500 gm possess velocities 10iˆ m/s and 3iˆ  5ˆj m/s respectively. The velocity of their centre of mass in m/s is 5ˆ 25 ˆ 5 i  25ˆj 5iˆ  j 25iˆ  ˆj (A) 5iˆ  25ˆj (B) (C) (D) 7 7 7 Where will be the centre of mass on combine two masses m and M (M > m) (A) Towards m (B) Towards M (C) Between m and M (D) Anywhere The centre of mass of a body (A) Lies always outside the body (B) May lie within, outside on the surface of the body (C) Lies always inside the body (D) Lies always on the surface of the body

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AIEEE

15.

Two bodies of masses 2 kg and 4 kg are moving with velocities 2 m/s and 10 m/s respectively along same direction. Then the velocity of their centre of mass will be (A) 8.1 m/s (B) 7.3 m/s (C) 6.4 m/s (D) 5.3 m/s

16.

A cricket bat is cut at the location of its centre of mass as shown. Then

17.

18.

(A) The two pieces will have the same mass (B) The bottom piece will have larger mass (C) The handle piece will have larger mass (D) Mass of handle piece is double the mass of bottom piece Ring of mass 6 kg and radius 40 cm is revolving at the rate of 300 revolutions per minute. Its moment of inertia be (A) 0.92 kg m2 (B) 0.96 kg m2 (C) 2.4 kg m2 (D) 2.98 kg m2 In Q. 17, the kinetic energy of rotation of the ring is 48 joule  The moment of inertia of a 500 gm cylinder of radius 10 cm about its natural axis is 0.5  10 3 kg m2 1.5  10 3 kg m2 (A) (B) (A)

19.

20.

48 2 joule

(B)

48 joule

(C)

48 joule

(D)

2  10 3 kg m2 2.5  103 kg m2 (C) (D) The angular momentum changes from 2 units to 6 units in 4 s. The torque is (A) 1 unit (B) 1/2 unit (C) 3/2 unit (D) 4 unit

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LEVEL - II 1. 2. 3. 4. 5.

6.

7.

8.

9. 10.

11.

12.

13.

A ring of mass 10 kg and diameter 0.4 m is rotated about its axis. If it makes 2100 revolutions per minute, then its angular momentum will be (A) 44 kg m2 s–1 (B) 88 kg m2 s–1 (C) 4.4 kg m2 s–1 (D) 0.4 kg m2 s–1 A wheel whose moment of inertia is 12 kg m 2 has an initial angular velocity of 40 rad/s. A constant torque of 20 Nm acts on the wheel. The time in which the wheel is accelerated to 100 rad/s is (A) 72 second (B) 16 second (C) 8 second (D) 36 second A constant couple of 400 Nm turns a wheel of moment of inertia 100 kgm 2 about an axis through its centre. The angular velocity gained in 4 second is (A) 12 rad s–1 (B) 16 rad s–1 (C) 20 rad s–1 (D) 24 rad s–1 Which of the following is conserved when the torque acting on a system is zero ? (A) K.E. (B) Angular momentum (C) Angular K.E. (D) Linear momentum A spacecraft of rotational inertia 105 kg m2 is having a spin of 0.1 rad s–1. The spacecraft, there is a rigidly mounted flywheel of moment of inertia 1000 kg m 2. The spin motion of the spacecraft can be stopped by changing the rotational rate of the flywheel by (A) 1 rad s–1 (B) 10 rad s–1 (C) 100 rad s–1 (D) 1000 rad s–1 A fan of moment of inertia 0.6 kg m2 is turned upto a working speed of 0.5 r.p.s. The angular momentum of the fan is  kg m2s 1 (A) 0.6 kg m2 s–1 (B) 6 kg m2 s–1 (C) 3 kg m2 s–1 (D) 6 A rigid body of moment of inertia  has an angular acceleration . If the power supplied to the body is P, its instantaneous angular velocity is (A) P (B) P/ (C) P/ (D) P/ The moment of inertia of a thin uniform rod about an axis passing through its centre and perpendicular to its length is 0. What is the M.I. about an axis through one end and perpendicular to the rod? (A) 1/2 0 (B) 30 (C) 50 (D) 40 A rod of length 1 m and mass 1/2 kg rotates at angular speed 6 rad s –1 about one of its ends. The kinetic energy of the rod is (A) 1 joule (B) 2 joule (C) 3 joule (D) 4 joule A sphere rolls down on an inclined plane of inclination . What is the acceleration as the sphere reaches bottom 5 3 2 2 gsin  gsin  gsin  gsin  (A) (B) (C) (D) 7 5 7 5 A solid cylinder of mass M and radius R rolls without slipping down an inclined plane of length L and height h. What is the speed of its centre of mass when the cylinder reaches its bottom? 3 4 4 gh 2 gh (A) (B) (C) (D) gh gh 4 3 A solid sphere (mass 2 M) and a thin hollow spherical shell (mass M) both of the same size, roll down an inclined plane, then (A) Solid sphere will reach the bottom first (B) Hollow spherical shell will reach the bottom first (C) Both will reach at the same time (D) None of these A solid sphere, a hollow sphere and a ring are released from top of an inclined plane (frictionless) so that they slide down the plane. Then maximum acceleration down the plane is for (no rolling) (A) Solid sphere (B) hollow sphere (C) Ring (D) All same

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14.

15.

16. 17.

18. 19.

20.

Chapter - 5 Rotational Motion

AIEEE

An inclined plane makes an angle of 30 with the horizontal. A solid sphere rolling down this inclined plane from rest without slipping has a linear acceleration equal to g 2g 5g 5g (A) (B) (C) (D) 3 3 7 14 A cord is wound round the circumference of wheel of radius r. The axis of the wheel is horizontal and moment of inertia about it is . A weight mg is attached to the end of the cord and falls from the rest. After falling through a distance h, the angular velocity of the wheel will be 1/ 2 1/ 2 2gh  2mgh  2mgh  2gh (A) (B) (C) (D)    mr 2    2mr 2   mr     Two particles A and B initially at rest move towards each other under a mutual force of attraction. At the instant when the speed of A is v and the speed of B is 2v, the speed of centre of mass of the system is (A) Zero (B) v (C) 1.5v (D) 3v A particle undergoes uniform circular motion. About which point on the plane of the circle, will the angular momentum of the particle remain conserved (A) Centre of the circle (B) On the circumference of the circle (C) Inside the circle (D) Outside the circle A mass of 2 kg is whirled in a horizontal circle by means of a string at an initial speed of 5 r.p.m. Keeping the radius constant, the tension in the string is doubled. The new speed is nearly (A) 14 r.p.m. (B) 10 r.p.m. (C) 20 r.p.m. (D) 7 r.p.m. A fly wheel rotates about a fixed axis and slows down from 300 r.p.m. to 100 r.p.m. in 2 minutes. Then its angular retardation in radian/minute2 is 100 100 200 (A) (B) 100 (C) (D)  A particle projected so as to just move along a vertical circle. The ratio of the tensions in the string when the particle is at the lowest and highest point on the circle is (A) 1 (B) finite but large (C) zero (D) infinite

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