Easy Problems

Easy Problems 1) A 4 kg block rests on a floor. The coefficient of static friction µs is 0.6. What horizontal force is required to move it? What force, acting at an angle 0 of 45 will be required to pull it? F

2) What force F is required to move this 4 kg block if it acts at an angle of 0

45 as shown? µs = 0.6. F

Is there an angle at which no matter how strong the applied force acts, the block will not move? 0

3) A 10 kg block is placed on an 45 incline. µs = 1.2. What force parallel to the incline is needed to start the block moving?

4) Given m = 12 kg, µk = 0.4, what is F so that the block moves at a constant velocity? F

5) The force required to keep a 10 kg block at constant speed is 5N. What is the coefficient of kinetic friction between it and the floor? 6) A box with weight 220 N rests on the floor. The coefficient of static friction between the box and the floor is 0.41 and the coefficient of kinetic friction is 0.32. a) What is the minimum magnitude of the horizontal force required to push it with to start it moving? b) Once moving, what is the minimum horizontal force required to keep it moving with constant velocity? c) If we continued to apply the force required to start it moving, 2 what would the acceleration be? Ans. a) 90 N b) 70 N c) 0.89 m/s . 7) A book is held against a vertical wall by a horizontal force as shown. µ (static) is 0.8. If the mass of the book is 5 kg, what minimum force is required? What happens to the force of friction if we apply twice the required minimum force? 8) A horizontal force F of 12 N pushes a block weighing 5 N against a vertical wall The coefficient of static friction µs between the wall and the block is 0.6 and the coefficient of kinetic friction µ k is 0.4. Assume that the block is not moving initially. Will it start moving? What is the force exerted by the block on the wall? y x Ans. No. 12i + 5j.

9) A 110g disk slides on the ice for 15 m before it stops. a) If its initial speed was 6.0 m/s, what is the coefficient of kinetic friction between disk and ice? Ans. 0.12. 10) A 3.5 kg block is pushed along the floor by a force F = 15N that makes 0 an angle of 40 with the floor. The coefficient of kinetic friction is 0.25. What is the magnitude of the frictional force and the acceleration of 2 the block? 11 N, rightward, 0.14 m/s .

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11) The block weighs 80 N. Incline is 20 . µs = 0.25, µ k = 0.15. What is the minimum force F along the incline as shown, that will prevent the block from slipping? What is the minimum F that will start the block moving up the incline? What value of F is required to keep the block moving up the incline at constant speed?

Ans. 8.6 N, 46 N, 39 N. 12) A box is pulled along a floor with constant velocity by exerting a force 0 on it at an angle of 45 with the vertical. If the box weighs 750 N, and the coefficient of kinetic friction is 0.15, what is the force used?

Ans. 138 N. 0

13) A body slides up an inclined plane forming an angle of 45 with the horizontal with an acceleration of g√2 down along the incline. If its mass is m, what is the force of friction acting on it? Ans. Mg/√2 down along the incline. 0

14) A 2 kg block slides down a 30 incline at a constant velocity. What is the coefficient of kinetic friction? 15) A 3 kg block is projected up an incline with the initial speed of 8 m/s as shown:

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The incline makes a 45 angle with the horizontal. If the coeff of kinetic friction is 0.3, and the coefficient of static friction 0.6, then how far up along the incline does it move before coming to a stop? How long did it take to reach this point? 16) In the problem above, does the block slide back down? Or does it stay put once it comes to rest ? If it slides back down, how much time does it take to come back down to its starting point? What is its final velocity? 17)

Two blocks, A and B, are tied to each other via a string as shown. The top 0 block is 8 kg and the lower one is 4 kg. They slide down a 30 incline. If the coefficient of kinetic friction between the pairs of surfaces is 0.3, what is the acceleration of the blocks and what is the tension in the string? Do the problem for the case that the blocks are placed with the 8 kg block below the other. 18) The masses are 1 kg each, the angle of the incline with the horizontal is 0 30 and the surfaces are frictionless. Find the acceleration of the blocks and the tension in the string.

2

Ans.2.45 m/s , 7.35 N. 0

19) A 2 kg body slides over an inclined plane forming an angle of 45 with the horizon. The relationship between the distance s traveled by the body 2 2 and the time t is described by the equation s = C t , where C = 1.73 m/s . What is the force of friction acting on the body? Ans. 10/√2 N. 20) The masses are 1 kg each and the inclined planes form angles of 45 0 and 30 with the horizontal. What is the acceleration of the blocks?

0

2

Ans.1.02 m/s . 0

21) A 2 kg block is given an initial velocity of 10 m/s up along a 45 incline. µk = 0.4, µs = 0.6 . How long does it slide up? Does it slide back down? If so, how long does it take to come back to its starting point?

22) The coefficient of kinetic friction is 0.2. What is the acceleration of the block if it is a) sliding down the slope and b) it has been given an upward 0 shove and it is still sliding up the slope? The makes 60 with the horizon. 2 2 Ans. a)(5√3 - 1) m/s down the incline. b)(5√3 + 1) m/s down the incline.

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23) A body slides over an inclined plane forming an angle of 45 with the horizon. After a distance of 36.4 cm the body acquires a velocity of 2 m/s. What is the coefficient of friction between the body and the incline? Ans. 0.2. 24) Equal weights, 1 kg each. No friction on the side of the table. Coefficient of friction between the block on top and the table top is 0.1. Find the acceleration of the weights and the tension in the cord. Ans. 4.4 2 m/s , 5.4 N.

Moderately difficult problems 25) The lower block has the mass 20 kg and the upper block the mass 10 kg. Friction between the blocks is determined by µs = 0.4, µk = 0.2. The floor is frictionless.

What is the maximum force that can be applied to the top block so that the blocks move as one? What is the maximum force that can be applied to the bottom block so that they move as one? 26) The same problem as above, except that now there is friction between the bottom block and the floor: µk = 0.1. 27) Two blocks sit on top of one another as shown. The coefficient of static friction between the top 2 kg block and the bottom 6 kg block is 0.6. There is no friction between the bottom block and the floor. A 10 N force is applied to the top block as shown. What happens? What is the force exerted by the bottom block on the top block? 28)In the above problem the applied force is increased to 20 N. If the coefficient of kinetic friction is 0.5, then what are the initial accelerations of the blocks? 29) A force of 20 N is applied to the block on the left as shown. The block on the left has a mass of 2 kg and a µk = 0.1 with the floor. The block on the right has a mass 4 kg and a µk = 0.2 with the floor. What is their acceleration and what is the force between the blocks? 30) A 3 kg body is lowered onto a conveyer belt moving at a constant velocity of 2 m/s. The coefficient of kinetic friction between the body and the belt is 0.2. How far along the belt will the body slide before it stops wrt the belt (that is it is finally moving with the belt at the same speed).

31)

40 30

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The masses are in equilibrium. The masses shown are in kgs. If the coefficient of static friction between the 4 kg mass and the table is 1.0, what is the force of friction? What is the maximum mass that can hang in equilibrium?

32) A coin is placed on a phonograph record on a turntable, at a distance of 5 cm. The turntable is now set in rotation so it rotates at 78 rpm. The coin stays at the same point on the record. What can you say about the coefficient of friction between the coin and the record surface?

33)

What are the minimum and maximum accelerations that the wedge can be given to the right if the block on it were not to move? µ s = 0.2, µ k = 0.1. What if there were no friction?

34) A road is banked at an angle θ so that cars moving at 40 km / h can round a turn of radius of curvature 100 m. What is θ? If a car were to negotiate this curve at a speed of 20 km / h, what should the minimum coefficient of static friction between the wheels and the ground be so the car does not slip? 35) A car of mass 400 kg goes around a turn of radius of curvature 10 m. If µs = 0.6 between the tyres and the road, what is the maximum speed with which the car can safely negotiate the turn?

36) A 10 kg box rests on the bed of a truck. If µs = 0.5, what is the maximum acceleration that the truck can have without the box sliding? If µk = 0.4, and the truck starts from rest and maintains a steady 2 acceleration of 8 m/s , how long will the box take to slide 2m?

37) What minimum force F is required to hold a 2 kg book against a wall if µs = 0.6 ? What is the value of friction if 2F were exerted against the book? F 45

38) With what minimum acceleration must a train move so that a 20 kg block placed against its front surface does not slip down? µs = 0.4.

What will happen if the train were to move with a larger acceleration? What is the value of the frictional force under these conditions? 39) For what range of accelerations will the 10 kg box shown not slide? The front surface of the train has a slope of –2. µs = 0.4.

40) A 6 kg box and a 2 kg box are connected by a string passing over a 0 0 0 pulley on a 60 -30 -90 wedge as shown. If µ = 0.2, what is the acceleration of the boxes?

41) A road with a radius of curvature 10m is banked so that vehicles moving at 10 m/s can safely negotiate the turn. If µs = 0.4, what are the maximum and minimum speeds with which vehicles can safely make the turn? 42) A 5 kg block rests on top of a 10 kg block as shown. When a force of 20 N is applied to the lower block, they move as one, with no relative sliding. If there is no friction between the floor and the lower block, what is the force of friction on the top block? Ans. 20/3 N. 43) A balloon of mass 1600 kg is descending at constant speed. How much mass must be thrown out if it is to ascend at constant speed? Assume that the buoyancy force acting on it is (1200 x 9.8 ) N and that the force of air friction is taken to be of the same magnitude when it goes up and down. Ans. 800 kg. 44) What is the minimum force F that has to be applied to the 4 kg block so that it does not slide down the side of the bigger 10 kg block? µs = 0.4 between the blocks. The floor is frictionless. F

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45) A 2 kg block slides down a 30 incline inside an elevator that has a 2 downward acceleration of 4 m/s . If the incline is 2m long and the block starts from rest in the elevator frame, how long does it take to reach the bottom of the incline?

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46) A 2 kg block rests on the inclined surface of a 45 wedge of mass 10 kg. µs = 1.2 between the block and the wedge. What is the maximum force F with which you can pull the wedge so that the block does not slide down the incline? The floor is frictionless.

47) What is the angular velocity with which a cylindrical drum must rotate about its axis if a block is to stay stuck to its side without slipping down? µs = 0.6. R = 2m.

48) The blocks A,B and C weigh 3 kg, 4kg and 8 kg respectively. The coefficient of sliding friction between any two surfaces is 0.25. A is held at rest by a massless rigid rod fixed to the wall, while B and C are connected by a massless cord passing over a fixed massless pulley. Find the force P needed to pull C at a constant speed. (IIT 78)

A B C

P

49) Two blocks A and B are connected by a string and a spring. The string passes over a frictionless pulley fixed to a stationary block C. B

A

The coefficient of sliding friction between the surfaces is 0.2, the mass of A is 2 kg, the masses are moving with uniform speed. What is the mass of B and what is the energy stored in the spring? k = 1960 N/m. (IIT 82) 50) A spinning sphere is placed at the corner shown. Its mass is 4 kg. What are the frictional forces acting on it if µ between the surfaces is 0.3?

30

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Difficult problems 51) A 2 kg block slides down an incline onto a conveyor belt which is moving at 2 m/s. If the block reaches the belt at a speed of 6 m/s and µ = 0.4, how far will it slide along the belt before it stops sliding with respect to the belt?

If the blocks are being released at the top every 2 seconds, how far apart are they at the far right end of the belt, when they have stopped sliding? 0

52) The angle of the incline is 60 . The block on the incline is 10 kg, has µk = 0.4 with wedge. The wedge has the mass 20 kg. The vertically hanging mass is 4 kg, has the same kinetic friction coefficient with the side of the wedge. What is the acceleration of the wedge?

53) A 40 N force is applied to the top block. What are the accelerations of the blocks? From the bottom up, the blocks have the masses 4 kg, 2 kg and 1kg. The coefficient of static friction between the blocks is 0.2 and the kinetic coefficient is 0.1. There is no friction between the floor and the bottom block.

54) A spinning cylinder of mass 6 kg is placed against a corner as shown. The coefficient of friction between it and the wall (or floor) is 0.2. What are the frictional forces experienced by the cylinder?

55) A block of mass m slides back and forth without friction from one end to the other of a semi-circular cut in a square wedge.

If the wedge has the mass M, what is the minimum static friction coefficient between wedge and floor so that it does not move? 56) The pulleys and ropes are massless. The block on the incline weighs 8 0 kg and the one on the floor 2 kg. The 30 wedge weighs 10 kg. What is

the acceleration of the 2 kg block if a) all surfaces are frictionless b) if the coefficient of friction between all pairs of surfaces is 0.2?

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57) A circular tube of radius 2 cm is held at an angle of inclination of 30 . A 1 kg cylinder of the same radius slides down inside it. If µk = 0.2, how long does it take to slide 20 cm? For simplicity, assume that the pressure exerted by the cylinder on the part of the tube that supports its weight is uniform

58) A hollow drum in the shape of a frustrum rotates about its axis with the minimum angular velocity ω1 required to keep a block of mass 1 kg from 0 sliding down its side. µs = 0.4. The wall makes an angle of 30 with the vertical. The block is initially at a level where the radius is 2m. At what angular velocity ω2 will it start to slide up? If it is disturbed so that it starts to slide if the angular velocity is maintained at ω2, then how long will it take to slide up to the level where R = 4m? µk = 0.2.

59) A rope of length L and mass m lies on a table with just the length of rope hanging in order to cause the rope to slide from the table. µ is given. What fraction of the length hangs from the table at this point?

How long will it take the entire rope to slide off the table? Make any reasonable assumptions. 60) A 5 kg weight and a 10 kg weight hang from a pulley. The pulley is weightless, the rope is massless, the coefficient of static friction between rope and pulley is 0.4. What is the acceleration of the blocks?

Assessment 1) The jaws of a vice clamp down on and exert 200N of force each on a rectangular copper block. If the coefficient of static friction between the block's surface and the jaw's surface is 0.6, what force must I apply to pull the block out? Neglect the weight of the block.

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2) A reluctant cow, of mass 400 kg, stands on hoofs that are 25 cm in area each. If the coefficient of static friction between the hoofs and the floor is 0.6, with what force would you have to tug on its tail to make it moove? 0

3) A 4 kg block slides down a 30 wedge which is accelerating to the right as 2 shown at 5 m/s . If the block started from rest wrt the wedge, how long will it take to slide 1m along the incline? µ = 0.2.

4) A 40 N force is applied to the middle block as shown. If the coefficient of static friction between the blocks is 0.3 and the coeff of kinetic friction is 0.2, what happens? The blocks are initially at rest. The masses shown are in kg, the floor is frictionless. 1 2 3 5) A 60 kg student climbs a ladder that is leaning against a wall at an angle 0 of 45 . The wall is frictionless, µs between ladder and floor is 0.4. At what height is the student when the ladder starts slipping? The ladder is massless.