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EXERCISE # 1 Track Questions based on

Q.1

40 m/s

A point moves along a circle with velocity v = at where a = 0.5 m/sec2. Then the total acceleration of the point at the moment when it covered (1/10)th of the circle after beginning of motion-

Q.2

30 m 30º T.V Camera

(A) 0.5 m/sec2

(B) 0.6 m/sec2

(A) 4/3 rad/sec

(B) 3/4 rad/sec

(C) 0.7 m/sec2

(D) 0.8 m/sec2

(C) 8/3 3 rad/sec

(D) 1 rad/sec

Angular position of a line of a disc of radius

Q.6

r = 6 cm is given by = 10 – 5t + 4t2 rad. the average angular speed between 1 and 3 s is-

Q.3

(A) rad/s

(B) 11 rad/s

(C) 22 rad/s

(D) 5.5 rad/s

Questions based on

Two moving particles P and Q are 10 m apart at a certain instant. The velocity of P is 8m/s making an angle 30º with the line joining P and Q and that of Q is 6m/s making an angle 30º with PQ as shown in the figure. Then angular velocity of P with respect to Q is6 m/s

A car is moving in a circular path of radius 500m with a speed of 30m/sec. If its speed is

10 m

increasing at the rate of 2m/sec2, the resultant acceleration will be -

P

(A) 2 m/sec2

8 m/s (A) Zero (C) 0.4 rad/sec

(C) 2.7 Q.4

car

Kinematics of circular motion

m/sec2

(B) 2.5 m/sec2 (D) 4

m/sec2

An electric fan has blades of length 30 cm as measured from the axis of rotation. If the fan is rotating at 1200 r.p.m. The acceleration of a point on the tip of the blade is about(A) 1600 m/sec2

(B) 4740 m/sec2

(C) 2370 m/sec2

(D) 5055 m/sec2

Questions based on

Q.7

General motion along curved path

30º

30º Q

(B) 0.1 rad/sec (D) 0.7 rad/sec

Dynamics of circular motion A particle of mass m is fixed to one end of a light spring of force constant k and unstretched length . The system is rotated about the other end of the spring with an angular velocity , in gravity free space. The increase in length of the spring will be

k Q.5

A racing car is travelling along a track at a constant speed of 40 m/s. A T.V. camera men is recording the event from a distance of 30m directly away from the track as shown in figure. In order to keep the car under view in the position shown, the angular speed with which the camera should be rotated, is-

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m

(A)

m2 k

(B)

(C)

m2 k m2

(D) None of these

m2 k m2

CIRCULAR MOTION

18

Q.8

A railway track is banked for a speed v, by making the height of the outer rail (h) higher than that of the inner rail. The distance between the rails is d. The radius of curvature of the track is r(A)

mv2 towards centre R mv2 (B) away from centre R mv2 (C) along tangent R (D) zero

(A)

h v2 = d rg

h v2 (B) tan sin 1 = d rg 2 h v (C) tan–1 = d rg

(D)

move in a circle of radius r with a uniform speed v. The centrifugal force on it is-

Q.12

h v2 = r dg

A car moves at a constant speed on a road as shown in figure. The normal force by the road on the car is NA and NB when it is at the points A and B A

Q.9

The tube AC forms a quarter circle in a vertical plane. The ball B has an area of cross-section slightly smaller than that of the tube, and can move without friction through it. B is placed at A and displaced slightly. It willA B

(A) NA = NB (B) NA > NB (C) NA < NB (D) insufficient information Q.13

C (A) always be in contact with the inner wall of the tube (B) always be in contact with the outer wall of the tube (C) initially be in contact with the inner wall and later with the outer wall (D) initially be in contact with the outer wall and later with the inner wall

Q.10

Q.11

A particle is acted upon by a constant force always normal to the direction of motion of the particle. It is therefore inferred that(i) Its velocity is constant (ii) It moves in a straight line (iii) Its speed is constant (iv) It moves in circular path (A) i , iv (B) iii, iv (C) i, ii (D) i, ii, iii

B

Three identical cars A, B and C are moving at the same speed on three bridges. The car A goes on plane bridge. B on a bridge convex upwards and car C on a bridge concave upwards. Let FA, FB and FC be the normal forces exerted by the cars on the bridges when they are at the middle of bridge (A) FA is maximum (B) FB is maximum (C) FC is maximum (D) FA = FB = FC

Q.14

A particle of mass m is observed from an inertial frame of reference and is found to

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A cylindrical bucket filled with water is whirled around in a vertical circle of radius r. What can be the minimum speed at the top of the path if water does not fall out from the bucket. If it continues with this speed. What normal force the bucket exerts on water at the lowest point of path ? (A)

Rg , 2mg

(B)

2Rg , 2mg CIRCULAR MOTION

19

(C)

3Rg , 3mg

(D)

Rg , 3mg

Q.15

A bucket tied at the end of a 1.6 m long string is whirled in a vertical circle with constant speed. What should be the minimum speed so that the water from the bucket does not spill, When the bucket is at the highest position (Take g = 10 m/sec2) (A) 4 m/sec (B) 6.25 m/sec (C) 16 m/sec (D) None of the above

Q.16

A smooth hollow cone whose vertical angle is 2, with it axis vertical and vertex downwards, revolves about its axis n times per second. Find distances from axis of rotation where a particle may be placed on the inner surface of cone so that it rotates with same speed g sin gcot (A) (B) 2 2 4 n 4 2 n 2 (C)

4 2 n 2 g

(D)

Q.19

The vertical section of a road over a canal bridge in the direction of its length is in the form of circle of radius 8.9 metre, Then the greatest speed at which the car can cross this bridge without losing contact with the road at its highest point, the centre of gravity of the car being at a height h = 1.1 metre from the ground is(Take g = 10 m/sec2 ) (A) 5 m/sec (B) 10 m/sec (C) 15 m/sec (D) 20 m/sec

Fill in the blanks type question Q.20

A particle is projected with a speed u at an angle with the horizontal. Considering a small part of its path near the highest position, the approximate radius of curvature is given by R =..............................

gsin 4 2 n 2

Q.17

A 2 kg stone at the end of a string 1m long is whirled in a vertical circle at a constant speed. The speed of the stone is 4 m/sec. The tension in the string will be 52N. when the stone is(A) At the top of the circle (B) At the bottom of the circle (C) Half way down (D) None of the above

Q18

A particle, moving equal magnitudes acceleration. The meters) (A) 2 (C)

along a circular path has of linear and angular diameter of path is (in (B) 1 (D) 2

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CIRCULAR MOTION

20

EXERCISE # 2 Part-A Q.1

(A) tan–1(v0/r) in the plane perpendicular

(Only single correct answer type questions)

to r (B) tan–1(r/v0) in the plane perpendicular

A car moves round a turn of constant curvature between A and B (curve AB = 100m) with a steady speed 72 km/hr. If angle between tangent at point A and B is 45º then magnitude of acceleration of car between A and B is -

A

B

100 m

to r

(C) tan–1 (r/v0) in the plane through r (D) Varying with time Q.4

A coin placed on a rotating turntable just slips if it is placed at a distance of 4 cm from the centre. If the angular velocity of the turntable is doubled, it will just slip at a distance of (A) 1 cm (B) 2 cm (C) 4 cm (D) 8 cm

Q.5

A simple pendulum having a bob of mass m is suspended from the ceiling of a car used in a stunt film shooting. The car moves up along an inclined cliff at a speed v and makes a jump to leave the cliff and lands at some distance. Let R be the maximum height of the car from the top of cliff. The tension in the string when the car is in air is -

45º

(B) 3.14 m/s2 (D) 6.28 m/s2

(A) Zero (C) 31.4 m/s2 Q.2

A simple pendulum is vibrating with an angular amplitude of 90º as shown in the following figure . For what value of is the acceleration directed O B B

(B) mg –

(A) mg

C A (i) Vertically upwards (ii) Horizontally (iii) Vertically downwards (A) 0º,cos–1(1/ 3 ),90º

(C) mg +

Q.6

(B) 90º,cos–1(1/ 3 ),0º (C) cos–1(1/ 3 ),0º,90º

mv2 R

B An open merry-go-round rotates at an angular velocity . A person stands in it at a distance r from the rotational axis. It is raining and raindrops fall vertically with a velocity v0. The person should hold an umbrella to protect himself with axis of umbrella tilted with vertical at angleCAREER POINT, CP Tower, Road No.1, IPIA, Kota (Raj.), Ph: 0744-3040000

(D) zero

A rod of length slides down along the inclined wall as shown in figure. At the instant when the speed of end A is v, speed of B is-

(D) cos–1(1/ 3 ), 90º,0º Q.3

mv2 R

y

x

v cos (A) cos

(C)

v sin cos

A v sin (B) sin (D)

v cos cos

CIRCULAR MOTION

21

Q.7

A stone of mass 1kg tied to a light inextensible string of length 10/3 metre is whirling in a vertical circle. If the ratio of maximum tension to minimum tension in the string is 4, then speed of stone at highest point of the circle is(A) 20 m/s

[g = 10 m/s2] (B) 103 m/s

(C) 52 m/s

(D) 10 m/s

Q.10

Q.11 Q.8

A small bead of mass m = 1 kg is carried by a circular hoop having centre at C and radius r = 1m which rotates about a fixed vertical axis. The coefficient of friction between bead and hoop is = 0.5. The maximum angular speed of the hoop for which the bead does not have relative motion with respect to hoop.

(D) 2 sin = 3 cos A stone is thrown horizontally with a velocity of 10m/sec. Find the radius of curvature of it's trajectory at the end of 3 sec after motion began (A) 10 10 m

(B) 100 10 m

(C) 10 m

(D) 100 m

A bead of mass m is located on a parabolic wire with its axis vertical and vertex at the origin as shown in figure and whose equation is x2 = 4ay. The wire frame is fixed and the bead can slide on it without friction. The bead is released from the point y = 4a on the wire frame from rest. The tangential acceleration of the bead when it reaches the position given by y = a is y m

C

x

O

45º m

(A) Q.12 1/ 2

Q.9

1/ 2

(A) (5 2 )

(B) (10 2 )

(C) (15 2 )1/ 2

(D) (30 2 )1/ 2

A particle initially at rest starts moving from point A on the surface of a fixed smooth hemisphere of radius r as shown. The particle looses its contact with hemisphere at point B. C is centre of the hemisphere. The equation relating and is A B r C (A) 3 sin = 2 cos (B) 2 sin = 3 cos (C) 3 sin = 2 cos

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g 2

(B)

3g 2

(C)

g

(D)

2

g 5

A disc of radius R has a light pole fixed perpendicular to the disc at the circumference which in turn has a pendulum of length R attached to its other end as shown in figure. The disc is rotated with a constant angular velocity . The string is making an angle 30º with the rod. Then the angular velocity of disc is R 30º R

1/ 2

3g (A) R

1/ 2

3g (B) 2R

CIRCULAR MOTION

22

1/ 2

Q.13

1/ 2

2g g (C) (D) 3 3R 3R Two cars having masses m1 and m2 move in

(C) mg <

Part-B

circles of radius r1 and r2. If they complete the circle in equal time. The ratio of their

Q.14

Q.15

m1 m2

(B)

r1 r2

(C)

Q.20

The position vector of a particle in a circular motion about the origin sweeps out equal area in equal times(A) velocity remains constant (B) speed remains constant (C) acceleration remains constant (D) tangential acceleration remains constant

Q.21

A car of mass M is moving on a horizontal circular path of radius r. At an instant its speed is v and is increasing at a rate a (A) the acceleration of the car is towards the centre of the path (B) the magnitude of the frictional force on the car is greater than mv2/R (C) the friction coefficient between the ground and the car is not less than a/g (D) the friction coefficient between the ground and the car is µ = tan–1v2/Rg

Q.22

A circular road of radius r is banked for a speed of v = 40 km/h. A car of mass m attempts to go on the circular road. The friction coefficient between the tyre and the road is negligible. Then(A) the car cannot make a turn without skidding (B) if the car turn at a speed less than 40 km/h, it will slip down. (C) if the car turn at the correct speed of 40 km/h the force by the road on the car is equal to mv2/r

A stone is moved round a horizontal circle with a 20 cm long string tied to it. If centripetal acceleration is 9.8 m/s2, then its angular velocity will be(A) 7 rad/s (B) 22/7 rad/s (C) 49 rad/s (D) 14 rad/s

Q.16

A motorcycle is going on an over bridge of radius R. The driver maintain a constant speed. As the motorcycle is ascending on the over bridge, the normal force on it (A) increases (B) decreases (C) remains same (D) fluctuates

Q.17

Let denote the angular displacement of a simple pendulum oscillating in a vertical plane. If the mass of the bob is m. The tension in the string is mg cos (A) always (B) never (C) at extreme position (D) at mean position

Q.18

Water in a bucket is whirled in a vertical circle with a string attached to it. The water does not fall down even when the bucket is inverted at the top of its path. We conclude that(A) mg =

mv2 R

(B) mg >

One or more than one correct answer type questions

An object follows a curved path. The following quantities may remain constant during the motion(A) speed (B) velocity (C) acceleration (D) magnitude of acceleration

m1r1 (D) 1 m 2 r2

A 30 cm diameter turn table starts from rest and takes 2 s to reach its final rotation rate of 33.5 rpm; the angular acceleration is(A) 1.75 rad/s2 (B) 1.25 rad/s2 (C) 2 rad/s2 (D) 1 rad/s2

(D) none of these

Q.19

angular speeds 1/2 is(A)

mv2 R

mv2 R

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CIRCULAR MOTION

23

(D) if the car turn at the correct speed of 40 km/h, the force by the road on the car is greater than mg as well as greater than mv2/r Q.23

are to choose any one of the following four responses. (A) If both Assertion and Reason are true and

A person applies a constant force F on a

the Reason is correct explanation of the

particle of mass m and finds that the particle

Assertion. (B) If both Assertion and Reason are true but

move in a circle of radius r with a uniform speed v.

Reason is not correct explanation of the

(A) this is not possible

Assertion.

(B) there are other forces also on the particle

(C) If Assertion is true but the Reason is false.

(C) the resultant of other forces is mv2/r

(D) If Assertion is false but Reason is true.

towards centre (D) the resultant of the other forces varies in

Q.25

magnitude as well as direction

Assertion : On a banked curved track, vertical components of normal reaction provides the necessary centripetal force.

Q.24

Figure shows a rod of length L pivoted near

Reason : Centripetal force is always required

an end and which is made to rotate in a

for turning.

horizontal plane with a constant angular speed. A ball of mass m is suspended by a

Q.26

Assertion

:

The

tendency

of

string also of length L from the other end of

skidding/overturning is quadrupled, when a

the rod. If is the angle made by string with

cyclist doubles his speed of turning.

the vertical, then-

Reason : tan =

L

v2 Rg

becomes 4 times as v is doubled. L T m

Part-D Column matching type questions Q.27

(A) T sin =

m2L(1

+ sin)

(B) T cos = mg (C) tan =

A particle is suspended from a string of length 'R'. It is given a velocity u = 3 Rg . Match the following

L(1 sin ) g 2

(D) none of above

C D

Part-C Assertion-Reason type questions

B A

u

Column-I The following questions consists of two statements each, printed as Assertion and

Column-II

(A) Velocity at B

(P) 7 mg

(B) Velocity at C

(Q)

5gR

Reason. While answering these questions you CAREER POINT, CP Tower, Road No.1, IPIA, Kota (Raj.), Ph: 0744-3040000

CIRCULAR MOTION

24

(C) Tension at B

(R)

(D) Tension at C

(S) 4 mg

Q.28

7gR

The bob of a simple pendulum is given a velocity 10 m/s at its lowest point. Mass of the bob is 1 kg and string length is 1 m. Column – I (A) Minimum tension in string (in Newton) (B) Magnitude of acceleration of bob when the string is horizontal (in m/s2)

Q.29

Column - II (P) 50

road of radius 35/ 3 . Angle of banking of road is 30º. Coefficient of friction between road and tyres is =

1 2 3

. Match the

following:

(Q) 60

Column-I

(C) Minimum magnitude of (R) zero acceleration of bob (in m/s2) (D) Tangential acceleration (S) 10

A car of mass 500 kg is moving in a circular

Column-II

(A) Maximum speed (in m/s) of

(P) 5 2

car for safe turning (B) Minimum speed (in m/s) of car for safe turning

(Q) 12.50

(C) Speed (in m/s) at which friction (R) 65

at the highest point (in m/s2)

210

force between tyres and road is zero (D) Friction force (in 102 Newton) (S)

350 3

between tyres and road if speed is

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350 m/s 6

CIRCULAR MOTION

25

EXERCISE # 3 Part-A Subjective Type Questions Q.1

Q.2

Q.7

A wet open umbrella is held upright and is rotated about the handle at a uniform rate of 21 revolutions in 44 s. If the rim of the umbrella is circle 1 metre in diameter and the height of the rim above the floor is 1.5 m, find where the drops of water spun off the rim and hit the floor. A man whirls a stone around his head on the end of a string 4 metre long. If the stone has a mass of 0.4 kg and the string will break if the tension in it exceeds 8 N, what is the smallest angle the string can make with the horizontal? What is the speed of the stone ? (g = 10 m/s2)

Q.3

A boy whirls a stone in a horizontal circle of radius 1.5 m and 2 m above the ground by means of a string. The string breaks and the stone flies off horizontally, striking the ground 10 m away. What is the centripetal acceleration during circular motion ?

Q.4

A stone is fastened to one end of a string and is whirled in a vertical circle of radius R. Find the minimum speed the stone can have at the highest point of the circle.

Q.5

A stone of mass 1 kg is attached to one end of a string of length 1 m and breaking strength 500 N, and is whirled in a horizontal circle on a frictionless table top. The other end of the string is kept fixed. Find the maximum speed the stone can attain without breaking the string.

Q.6

A spaceman in training is rotated in a seat at the end of horizontal rotating arm of length 5 m. If he can withstand acceleration up to 9 g, what is the maximum number of revolutions per second permissible ? Take g = 10 m/s2

An insect on the axle of a wheel observes the motion of a particle and 'finds' it to take its place along the circumference of a circle of radius 'R' with a uniform angular speed . The axle is moving with a uniform speed 'v' relative to the ground. How will an observer on the ground describe the motion of the same point.

Q. 8

A circular automobile test track has a radius of 200 m. The track is so designed that when a car travels at a speed of 100 kilometer per hour, the force between the automobile and the track is normal to the surface of track. Find the angle of the bank.

Q. 9

A cyclist is riding with a speed of 27 kmh–1. As he approaches a circular turn on the road of radius 80 m, he applies brakes and reduces his speed at the constant rate of 0.5 ms–1 every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn ?

Q.10

A motorcycle has to move with a constant speed on an over bridge which is in the form of a circular arc of radius R and has a total length L. Suppose the motorcycle starts from the highest point. (a) What can its maximum velocity be for which contact with road is not broken at the highest point ? (b) If the 1 motorcycle goes at speed times the 2 maximum found in part (a). Where will it lose the contact with the road ? (c) What maximum uniform speed can it maintain on the bridge if it does not lose contact anywhere on the bridge ?

Q.11

A stone is thrown horizontally with a velocity 10 m/s. Find the radius of curvature of its trajectory in 3 second after the motion began. Disregard the resistance of air.

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CIRCULAR MOTION

26

Q.12

Q.13

A simple pendulum is suspended from the ceiling of a car taking a turn of radius 10 m at a speed of 36 km/h. Find the angle made by the string of the pendulum with the vertical if this angle does not change during the turn. Take g = 10 m/s2.

B O

Q.17

A particle of mass m moves along the internal smooth surface of vertical cylinder of radius R. Find the force with which the particle acts on the cylinder wall if at the initial moment of time its velocity equals v0 and forms an angle

A heavy particle hanging from a fixed point projected horizontally with speed g , find

Q.15

Q.16

A hemispherical bowl of radius R is rotated about its axis of symmetry which is kept vertical. A small block is kept in the bowl at a position where the radius makes angle with the vertical. The block rotates with the bowl without any slipping. The frictional coefficient between the block and the bowl is µ. Find the range of angular speed for which the block will not slip. A table with smooth horizontal surface is fixed in a cabin that rotates with a uniform angular velocity in a circular path of radius R. A smooth groove AB of length L

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Q.1

40 m/s

A point moves along a circle with velocity v = at where a = 0.5 m/sec2. Then the total acceleration of the point at the moment when it covered (1/10)th of the circle after beginning of motion-

Q.2

30 m 30º T.V Camera

(A) 0.5 m/sec2

(B) 0.6 m/sec2

(A) 4/3 rad/sec

(B) 3/4 rad/sec

(C) 0.7 m/sec2

(D) 0.8 m/sec2

(C) 8/3 3 rad/sec

(D) 1 rad/sec

Angular position of a line of a disc of radius

Q.6

r = 6 cm is given by = 10 – 5t + 4t2 rad. the average angular speed between 1 and 3 s is-

Q.3

(A) rad/s

(B) 11 rad/s

(C) 22 rad/s

(D) 5.5 rad/s

Questions based on

Two moving particles P and Q are 10 m apart at a certain instant. The velocity of P is 8m/s making an angle 30º with the line joining P and Q and that of Q is 6m/s making an angle 30º with PQ as shown in the figure. Then angular velocity of P with respect to Q is6 m/s

A car is moving in a circular path of radius 500m with a speed of 30m/sec. If its speed is

10 m

increasing at the rate of 2m/sec2, the resultant acceleration will be -

P

(A) 2 m/sec2

8 m/s (A) Zero (C) 0.4 rad/sec

(C) 2.7 Q.4

car

Kinematics of circular motion

m/sec2

(B) 2.5 m/sec2 (D) 4

m/sec2

An electric fan has blades of length 30 cm as measured from the axis of rotation. If the fan is rotating at 1200 r.p.m. The acceleration of a point on the tip of the blade is about(A) 1600 m/sec2

(B) 4740 m/sec2

(C) 2370 m/sec2

(D) 5055 m/sec2

Questions based on

Q.7

General motion along curved path

30º

30º Q

(B) 0.1 rad/sec (D) 0.7 rad/sec

Dynamics of circular motion A particle of mass m is fixed to one end of a light spring of force constant k and unstretched length . The system is rotated about the other end of the spring with an angular velocity , in gravity free space. The increase in length of the spring will be

k Q.5

A racing car is travelling along a track at a constant speed of 40 m/s. A T.V. camera men is recording the event from a distance of 30m directly away from the track as shown in figure. In order to keep the car under view in the position shown, the angular speed with which the camera should be rotated, is-

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m

(A)

m2 k

(B)

(C)

m2 k m2

(D) None of these

m2 k m2

CIRCULAR MOTION

18

Q.8

A railway track is banked for a speed v, by making the height of the outer rail (h) higher than that of the inner rail. The distance between the rails is d. The radius of curvature of the track is r(A)

mv2 towards centre R mv2 (B) away from centre R mv2 (C) along tangent R (D) zero

(A)

h v2 = d rg

h v2 (B) tan sin 1 = d rg 2 h v (C) tan–1 = d rg

(D)

move in a circle of radius r with a uniform speed v. The centrifugal force on it is-

Q.12

h v2 = r dg

A car moves at a constant speed on a road as shown in figure. The normal force by the road on the car is NA and NB when it is at the points A and B A

Q.9

The tube AC forms a quarter circle in a vertical plane. The ball B has an area of cross-section slightly smaller than that of the tube, and can move without friction through it. B is placed at A and displaced slightly. It willA B

(A) NA = NB (B) NA > NB (C) NA < NB (D) insufficient information Q.13

C (A) always be in contact with the inner wall of the tube (B) always be in contact with the outer wall of the tube (C) initially be in contact with the inner wall and later with the outer wall (D) initially be in contact with the outer wall and later with the inner wall

Q.10

Q.11

A particle is acted upon by a constant force always normal to the direction of motion of the particle. It is therefore inferred that(i) Its velocity is constant (ii) It moves in a straight line (iii) Its speed is constant (iv) It moves in circular path (A) i , iv (B) iii, iv (C) i, ii (D) i, ii, iii

B

Three identical cars A, B and C are moving at the same speed on three bridges. The car A goes on plane bridge. B on a bridge convex upwards and car C on a bridge concave upwards. Let FA, FB and FC be the normal forces exerted by the cars on the bridges when they are at the middle of bridge (A) FA is maximum (B) FB is maximum (C) FC is maximum (D) FA = FB = FC

Q.14

A particle of mass m is observed from an inertial frame of reference and is found to

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A cylindrical bucket filled with water is whirled around in a vertical circle of radius r. What can be the minimum speed at the top of the path if water does not fall out from the bucket. If it continues with this speed. What normal force the bucket exerts on water at the lowest point of path ? (A)

Rg , 2mg

(B)

2Rg , 2mg CIRCULAR MOTION

19

(C)

3Rg , 3mg

(D)

Rg , 3mg

Q.15

A bucket tied at the end of a 1.6 m long string is whirled in a vertical circle with constant speed. What should be the minimum speed so that the water from the bucket does not spill, When the bucket is at the highest position (Take g = 10 m/sec2) (A) 4 m/sec (B) 6.25 m/sec (C) 16 m/sec (D) None of the above

Q.16

A smooth hollow cone whose vertical angle is 2, with it axis vertical and vertex downwards, revolves about its axis n times per second. Find distances from axis of rotation where a particle may be placed on the inner surface of cone so that it rotates with same speed g sin gcot (A) (B) 2 2 4 n 4 2 n 2 (C)

4 2 n 2 g

(D)

Q.19

The vertical section of a road over a canal bridge in the direction of its length is in the form of circle of radius 8.9 metre, Then the greatest speed at which the car can cross this bridge without losing contact with the road at its highest point, the centre of gravity of the car being at a height h = 1.1 metre from the ground is(Take g = 10 m/sec2 ) (A) 5 m/sec (B) 10 m/sec (C) 15 m/sec (D) 20 m/sec

Fill in the blanks type question Q.20

A particle is projected with a speed u at an angle with the horizontal. Considering a small part of its path near the highest position, the approximate radius of curvature is given by R =..............................

gsin 4 2 n 2

Q.17

A 2 kg stone at the end of a string 1m long is whirled in a vertical circle at a constant speed. The speed of the stone is 4 m/sec. The tension in the string will be 52N. when the stone is(A) At the top of the circle (B) At the bottom of the circle (C) Half way down (D) None of the above

Q18

A particle, moving equal magnitudes acceleration. The meters) (A) 2 (C)

along a circular path has of linear and angular diameter of path is (in (B) 1 (D) 2

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CIRCULAR MOTION

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EXERCISE # 2 Part-A Q.1

(A) tan–1(v0/r) in the plane perpendicular

(Only single correct answer type questions)

to r (B) tan–1(r/v0) in the plane perpendicular

A car moves round a turn of constant curvature between A and B (curve AB = 100m) with a steady speed 72 km/hr. If angle between tangent at point A and B is 45º then magnitude of acceleration of car between A and B is -

A

B

100 m

to r

(C) tan–1 (r/v0) in the plane through r (D) Varying with time Q.4

A coin placed on a rotating turntable just slips if it is placed at a distance of 4 cm from the centre. If the angular velocity of the turntable is doubled, it will just slip at a distance of (A) 1 cm (B) 2 cm (C) 4 cm (D) 8 cm

Q.5

A simple pendulum having a bob of mass m is suspended from the ceiling of a car used in a stunt film shooting. The car moves up along an inclined cliff at a speed v and makes a jump to leave the cliff and lands at some distance. Let R be the maximum height of the car from the top of cliff. The tension in the string when the car is in air is -

45º

(B) 3.14 m/s2 (D) 6.28 m/s2

(A) Zero (C) 31.4 m/s2 Q.2

A simple pendulum is vibrating with an angular amplitude of 90º as shown in the following figure . For what value of is the acceleration directed O B B

(B) mg –

(A) mg

C A (i) Vertically upwards (ii) Horizontally (iii) Vertically downwards (A) 0º,cos–1(1/ 3 ),90º

(C) mg +

Q.6

(B) 90º,cos–1(1/ 3 ),0º (C) cos–1(1/ 3 ),0º,90º

mv2 R

B An open merry-go-round rotates at an angular velocity . A person stands in it at a distance r from the rotational axis. It is raining and raindrops fall vertically with a velocity v0. The person should hold an umbrella to protect himself with axis of umbrella tilted with vertical at angleCAREER POINT, CP Tower, Road No.1, IPIA, Kota (Raj.), Ph: 0744-3040000

(D) zero

A rod of length slides down along the inclined wall as shown in figure. At the instant when the speed of end A is v, speed of B is-

(D) cos–1(1/ 3 ), 90º,0º Q.3

mv2 R

y

x

v cos (A) cos

(C)

v sin cos

A v sin (B) sin (D)

v cos cos

CIRCULAR MOTION

21

Q.7

A stone of mass 1kg tied to a light inextensible string of length 10/3 metre is whirling in a vertical circle. If the ratio of maximum tension to minimum tension in the string is 4, then speed of stone at highest point of the circle is(A) 20 m/s

[g = 10 m/s2] (B) 103 m/s

(C) 52 m/s

(D) 10 m/s

Q.10

Q.11 Q.8

A small bead of mass m = 1 kg is carried by a circular hoop having centre at C and radius r = 1m which rotates about a fixed vertical axis. The coefficient of friction between bead and hoop is = 0.5. The maximum angular speed of the hoop for which the bead does not have relative motion with respect to hoop.

(D) 2 sin = 3 cos A stone is thrown horizontally with a velocity of 10m/sec. Find the radius of curvature of it's trajectory at the end of 3 sec after motion began (A) 10 10 m

(B) 100 10 m

(C) 10 m

(D) 100 m

A bead of mass m is located on a parabolic wire with its axis vertical and vertex at the origin as shown in figure and whose equation is x2 = 4ay. The wire frame is fixed and the bead can slide on it without friction. The bead is released from the point y = 4a on the wire frame from rest. The tangential acceleration of the bead when it reaches the position given by y = a is y m

C

x

O

45º m

(A) Q.12 1/ 2

Q.9

1/ 2

(A) (5 2 )

(B) (10 2 )

(C) (15 2 )1/ 2

(D) (30 2 )1/ 2

A particle initially at rest starts moving from point A on the surface of a fixed smooth hemisphere of radius r as shown. The particle looses its contact with hemisphere at point B. C is centre of the hemisphere. The equation relating and is A B r C (A) 3 sin = 2 cos (B) 2 sin = 3 cos (C) 3 sin = 2 cos

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g 2

(B)

3g 2

(C)

g

(D)

2

g 5

A disc of radius R has a light pole fixed perpendicular to the disc at the circumference which in turn has a pendulum of length R attached to its other end as shown in figure. The disc is rotated with a constant angular velocity . The string is making an angle 30º with the rod. Then the angular velocity of disc is R 30º R

1/ 2

3g (A) R

1/ 2

3g (B) 2R

CIRCULAR MOTION

22

1/ 2

Q.13

1/ 2

2g g (C) (D) 3 3R 3R Two cars having masses m1 and m2 move in

(C) mg <

Part-B

circles of radius r1 and r2. If they complete the circle in equal time. The ratio of their

Q.14

Q.15

m1 m2

(B)

r1 r2

(C)

Q.20

The position vector of a particle in a circular motion about the origin sweeps out equal area in equal times(A) velocity remains constant (B) speed remains constant (C) acceleration remains constant (D) tangential acceleration remains constant

Q.21

A car of mass M is moving on a horizontal circular path of radius r. At an instant its speed is v and is increasing at a rate a (A) the acceleration of the car is towards the centre of the path (B) the magnitude of the frictional force on the car is greater than mv2/R (C) the friction coefficient between the ground and the car is not less than a/g (D) the friction coefficient between the ground and the car is µ = tan–1v2/Rg

Q.22

A circular road of radius r is banked for a speed of v = 40 km/h. A car of mass m attempts to go on the circular road. The friction coefficient between the tyre and the road is negligible. Then(A) the car cannot make a turn without skidding (B) if the car turn at a speed less than 40 km/h, it will slip down. (C) if the car turn at the correct speed of 40 km/h the force by the road on the car is equal to mv2/r

A stone is moved round a horizontal circle with a 20 cm long string tied to it. If centripetal acceleration is 9.8 m/s2, then its angular velocity will be(A) 7 rad/s (B) 22/7 rad/s (C) 49 rad/s (D) 14 rad/s

Q.16

A motorcycle is going on an over bridge of radius R. The driver maintain a constant speed. As the motorcycle is ascending on the over bridge, the normal force on it (A) increases (B) decreases (C) remains same (D) fluctuates

Q.17

Let denote the angular displacement of a simple pendulum oscillating in a vertical plane. If the mass of the bob is m. The tension in the string is mg cos (A) always (B) never (C) at extreme position (D) at mean position

Q.18

Water in a bucket is whirled in a vertical circle with a string attached to it. The water does not fall down even when the bucket is inverted at the top of its path. We conclude that(A) mg =

mv2 R

(B) mg >

One or more than one correct answer type questions

An object follows a curved path. The following quantities may remain constant during the motion(A) speed (B) velocity (C) acceleration (D) magnitude of acceleration

m1r1 (D) 1 m 2 r2

A 30 cm diameter turn table starts from rest and takes 2 s to reach its final rotation rate of 33.5 rpm; the angular acceleration is(A) 1.75 rad/s2 (B) 1.25 rad/s2 (C) 2 rad/s2 (D) 1 rad/s2

(D) none of these

Q.19

angular speeds 1/2 is(A)

mv2 R

mv2 R

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CIRCULAR MOTION

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(D) if the car turn at the correct speed of 40 km/h, the force by the road on the car is greater than mg as well as greater than mv2/r Q.23

are to choose any one of the following four responses. (A) If both Assertion and Reason are true and

A person applies a constant force F on a

the Reason is correct explanation of the

particle of mass m and finds that the particle

Assertion. (B) If both Assertion and Reason are true but

move in a circle of radius r with a uniform speed v.

Reason is not correct explanation of the

(A) this is not possible

Assertion.

(B) there are other forces also on the particle

(C) If Assertion is true but the Reason is false.

(C) the resultant of other forces is mv2/r

(D) If Assertion is false but Reason is true.

towards centre (D) the resultant of the other forces varies in

Q.25

magnitude as well as direction

Assertion : On a banked curved track, vertical components of normal reaction provides the necessary centripetal force.

Q.24

Figure shows a rod of length L pivoted near

Reason : Centripetal force is always required

an end and which is made to rotate in a

for turning.

horizontal plane with a constant angular speed. A ball of mass m is suspended by a

Q.26

Assertion

:

The

tendency

of

string also of length L from the other end of

skidding/overturning is quadrupled, when a

the rod. If is the angle made by string with

cyclist doubles his speed of turning.

the vertical, then-

Reason : tan =

L

v2 Rg

becomes 4 times as v is doubled. L T m

Part-D Column matching type questions Q.27

(A) T sin =

m2L(1

+ sin)

(B) T cos = mg (C) tan =

A particle is suspended from a string of length 'R'. It is given a velocity u = 3 Rg . Match the following

L(1 sin ) g 2

(D) none of above

C D

Part-C Assertion-Reason type questions

B A

u

Column-I The following questions consists of two statements each, printed as Assertion and

Column-II

(A) Velocity at B

(P) 7 mg

(B) Velocity at C

(Q)

5gR

Reason. While answering these questions you CAREER POINT, CP Tower, Road No.1, IPIA, Kota (Raj.), Ph: 0744-3040000

CIRCULAR MOTION

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(C) Tension at B

(R)

(D) Tension at C

(S) 4 mg

Q.28

7gR

The bob of a simple pendulum is given a velocity 10 m/s at its lowest point. Mass of the bob is 1 kg and string length is 1 m. Column – I (A) Minimum tension in string (in Newton) (B) Magnitude of acceleration of bob when the string is horizontal (in m/s2)

Q.29

Column - II (P) 50

road of radius 35/ 3 . Angle of banking of road is 30º. Coefficient of friction between road and tyres is =

1 2 3

. Match the

following:

(Q) 60

Column-I

(C) Minimum magnitude of (R) zero acceleration of bob (in m/s2) (D) Tangential acceleration (S) 10

A car of mass 500 kg is moving in a circular

Column-II

(A) Maximum speed (in m/s) of

(P) 5 2

car for safe turning (B) Minimum speed (in m/s) of car for safe turning

(Q) 12.50

(C) Speed (in m/s) at which friction (R) 65

at the highest point (in m/s2)

210

force between tyres and road is zero (D) Friction force (in 102 Newton) (S)

350 3

between tyres and road if speed is

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350 m/s 6

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EXERCISE # 3 Part-A Subjective Type Questions Q.1

Q.2

Q.7

A wet open umbrella is held upright and is rotated about the handle at a uniform rate of 21 revolutions in 44 s. If the rim of the umbrella is circle 1 metre in diameter and the height of the rim above the floor is 1.5 m, find where the drops of water spun off the rim and hit the floor. A man whirls a stone around his head on the end of a string 4 metre long. If the stone has a mass of 0.4 kg and the string will break if the tension in it exceeds 8 N, what is the smallest angle the string can make with the horizontal? What is the speed of the stone ? (g = 10 m/s2)

Q.3

A boy whirls a stone in a horizontal circle of radius 1.5 m and 2 m above the ground by means of a string. The string breaks and the stone flies off horizontally, striking the ground 10 m away. What is the centripetal acceleration during circular motion ?

Q.4

A stone is fastened to one end of a string and is whirled in a vertical circle of radius R. Find the minimum speed the stone can have at the highest point of the circle.

Q.5

A stone of mass 1 kg is attached to one end of a string of length 1 m and breaking strength 500 N, and is whirled in a horizontal circle on a frictionless table top. The other end of the string is kept fixed. Find the maximum speed the stone can attain without breaking the string.

Q.6

A spaceman in training is rotated in a seat at the end of horizontal rotating arm of length 5 m. If he can withstand acceleration up to 9 g, what is the maximum number of revolutions per second permissible ? Take g = 10 m/s2

An insect on the axle of a wheel observes the motion of a particle and 'finds' it to take its place along the circumference of a circle of radius 'R' with a uniform angular speed . The axle is moving with a uniform speed 'v' relative to the ground. How will an observer on the ground describe the motion of the same point.

Q. 8

A circular automobile test track has a radius of 200 m. The track is so designed that when a car travels at a speed of 100 kilometer per hour, the force between the automobile and the track is normal to the surface of track. Find the angle of the bank.

Q. 9

A cyclist is riding with a speed of 27 kmh–1. As he approaches a circular turn on the road of radius 80 m, he applies brakes and reduces his speed at the constant rate of 0.5 ms–1 every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn ?

Q.10

A motorcycle has to move with a constant speed on an over bridge which is in the form of a circular arc of radius R and has a total length L. Suppose the motorcycle starts from the highest point. (a) What can its maximum velocity be for which contact with road is not broken at the highest point ? (b) If the 1 motorcycle goes at speed times the 2 maximum found in part (a). Where will it lose the contact with the road ? (c) What maximum uniform speed can it maintain on the bridge if it does not lose contact anywhere on the bridge ?

Q.11

A stone is thrown horizontally with a velocity 10 m/s. Find the radius of curvature of its trajectory in 3 second after the motion began. Disregard the resistance of air.

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Q.12

Q.13

A simple pendulum is suspended from the ceiling of a car taking a turn of radius 10 m at a speed of 36 km/h. Find the angle made by the string of the pendulum with the vertical if this angle does not change during the turn. Take g = 10 m/s2.

B O

Q.17

A particle of mass m moves along the internal smooth surface of vertical cylinder of radius R. Find the force with which the particle acts on the cylinder wall if at the initial moment of time its velocity equals v0 and forms an angle

A heavy particle hanging from a fixed point projected horizontally with speed g , find

Q.15

Q.16

A hemispherical bowl of radius R is rotated about its axis of symmetry which is kept vertical. A small block is kept in the bowl at a position where the radius makes angle with the vertical. The block rotates with the bowl without any slipping. The frictional coefficient between the block and the bowl is µ. Find the range of angular speed for which the block will not slip. A table with smooth horizontal surface is fixed in a cabin that rotates with a uniform angular velocity in a circular path of radius R. A smooth groove AB of length L

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