Friction

A small box B of mass 2 kg is dragged in a straight line, along a rough horizontal plane, at a constant speed by a force of magnitude 5 N. The line of action of the force makes an angle α with the plane, where sinα = as shown in Figure 2.

Show that the magnitude of the normal reaction of the plane on the box is 16.6 N. At the instant when B is at the point O on the plane, the force of magnitude 5 N is removed.

Describe the motion of the box after the force of magnitude 5 N is removed.

Find the magnitude of the normal reaction of the plane on the box after the force of magnitude 5 N is removed. Given that after the force of magnitude 5 N is removed

• the box is modelled as a particle

• air resistance is modelled as being negligible

• the coefficient of friction between the box and the plane is modelled as 0.2

• the speed of the box as it passes through O is 4 m s^–1

• the box comes to rest at the point X on the plane

use the model to find the length OX.

State one limitation of the model, apart from ignoring air resistance, that could affect your answer to part (d).

A uniform rod AB has mass M and length 2a. A particle of mass 2M is attached to the rod at the point C, where AC = 0.5a The rod rests with end A on rough horizontal ground and end B against a vertical wall. The rod lies in a vertical plane which is perpendicular to the wall. The rod is in equilibrium at an angle α to the wall, as shown in Figure 4. In an initial model

• the vertical wall is modelled as being smooth

• the magnitude of the normal reaction of the ground on the rod at A is R

• the magnitude of the force exerted on the rod by the wall at B is S

Using the model,

find R in terms of M and g

show that S = Mg tanα In a refined model

• the vertical wall is modelled as being rough

• the magnitude of the normal reaction of the ground on the rod at A is R₁

State which is greater, R or R₁, giving a reason for your answer.

A second particle of mass 3M is now attached to the rod at B. The rod again rests with end A on rough horizontal ground and end B against the vertical wall. The rod lies in a vertical plane which is perpendicular to the wall. The rod is now in limiting equilibrium at an angle β to the wall, as shown in Figure 5. The vertical wall is again modelled as being smooth. The coefficient of friction between the rod and the ground is μ

use the model to show that μ =

TOTAL FOR MECHANICS IS 50 MARKS

Figure 1 shows a particle P of mass 0.5 kg at rest on a rough horizontal plane.

Find the magnitude of the normal reaction of the plane on P. The coefficient of friction between P and the plane is A horizontal force of magnitude X newtons is applied to P. Given that P is now in limiting equilibrium,

find the value of X.

A particle P of mass m is held at rest at a point on a rough inclined plane, as shown in Figure 3. It is given that

• the plane is inclined to the horizontal at an angle α, where tanα =

• the coefficient of friction between P and the plane is μ, whereμ

The particle P is released from rest and slides down the plane. Air resistance is modelled as being negligible. Using the model,

find, in terms of m and g, the magnitude of the normal reaction of the plane on P,

show that, as P slides down the plane, the acceleration of P down the plane is

State what would happen to P if it is released from rest butμ

Figure 5 shows a uniform rod AB of mass M and length 2a.

• the rod has its end A on rough horizontal ground

• the rod rests in equilibrium against a small smooth fixed horizontal peg P

• the point C on the rod, where AC = 1.5 a, is the point of contact between the rod and the peg

• the rod is at an angle θ to the ground, where tanθ =

The rod lies in a vertical plane perpendicular to the peg. The magnitude of the normal reaction of the peg on the rod at C is S.

The coefficient of friction between the rod and the ground is μ. Given that the rod is in limiting equilibrium,

find the value of μ.

TOTAL FOR MECHANICS IS 50 MARKS

A particle P has mass 5 kg. The particle is pulled along a rough horizontal plane by a horizontal force of magnitude 28 N. The only resistance to motion is a frictional force of magnitude F newtons, as shown in Figure 1.

Find the magnitude of the normal reaction of the plane on P The particle is accelerating along the plane at 1.4 m s^–2

Find the value of F The coefficient of friction between P and the plane is μ

Find the value of μ, giving your answer to 2 significant figures.

A rod AB has mass M and length 2a. The rod has its end A on rough horizontal ground and its end B against a smooth vertical wall. The rod makes an angle θ with the ground, as shown in Figure 3. The rod is at rest in limiting equilibrium.

State the direction (left or right on Figure 3 above) of the frictional force acting on the rod at A. Give a reason for your answer. The magnitude of the normal reaction of the wall on the rod at B is S. In an initial model, the rod is modelled as being uniform. Use this initial model to answer parts (b), (c) and (d).

By taking moments about A, show that

The coefficient of friction between the rod and the ground is μ

find the value of μ

find, in terms of M and g, the magnitude of the resultant force acting on the rod at A. In a new model, the rod is modelled as being non-uniform, with its centre of mass closer to B than it is to A. A new value for S is calculated using this new model, with tanθ =

State whether this new value for S is larger, smaller or equal to the value that S would take using the initial model. Give a reason for your answer.

TOTAL FOR MECHANICS IS 50 MARKS

A small stone A of mass 3m is attached to one end of a string.

A small stone B of mass m is attached to the other end of the string.

Initially A is held at rest on a fixed rough plane. The plane is inclined to the horizontal at an angle α, where tanα =

The string passes over a pulley P that is fixed at the top of the plane.

The part of the string from A to P is parallel to a line of greatest slope of the plane.

Stone B hangs freely below P, as shown in Figure 1. The coefficient of friction between A and the plane is

Stone A is released from rest and begins to move down the plane.

The stones are modelled as particles.

The pulley is modelled as being small and smooth.

The string is modelled as being light and inextensible.

Using the model for the motion of the system before B reaches the pulley,

write down an equation of motion for A

show that the acceleration of A is g

sketch a velocity-time graph for the motion of B, from the instant when A is released from rest to the instant just before B reaches the pulley, explaining your answer.

In reality, the string is not light.

State how this would affect the working in part (b).

A small stone A of mass 3m is attached to one end of a string. A small stone B of mass m is attached to the other end of the string. Initially A is held at rest on a fixed rough plane. The plane is inclined to the horizontal at an angle α, where tanα = The string passes over a pulley P that is fixed at the top of the plane. The part of the string from A to P is parallel to a line of greatest slope of the plane. Stone B hangs freely below P, as shown in Figure 1. The coefficient of friction between A and the plane is Stone A is released from rest and begins to move down the plane. The stones are modelled as particles. The pulley is modelled as being small and smooth. The string is modelled as being light and inextensible.

Using the model for the motion of the system before B reaches the pulley,

write down an equation of motion for A

show that the acceleration of A is g

sketch a velocity-time graph for the motion of B, from the instant when A is released from rest to the instant just before B reaches the pulley, explaining your answer. In reality, the string is not light.

State how this would affect the working in part (b).

A beam AB has mass m and length 2a.

The beam rests in equilibrium with A on rough horizontal ground and with B against a smooth vertical wall.

The beam is inclined to the horizontal at an angle θ, as shown in Figure 2.

The coefficient of friction between the beam and the ground is μ

The beam is modelled as a uniform rod resting in a vertical plane that is perpendicular to the wall.

Using the model,

A horizontal force of magnitude kmg, where k is a constant, is now applied to the beam at A.

This force acts in a direction that is perpendicular to the wall and towards the wall. Given that tanθ = , μ = and the beam is now in limiting equilibrium,

use the model to find the value of k.

A beam AB has mass m and length 2a. The beam rests in equilibrium with A on rough horizontal ground and with B against a smooth vertical wall. The beam is inclined to the horizontal at an angle θ, as shown in Figure 2. The coefficient of friction between the beam and the ground is μ The beam is modelled as a uniform rod resting in a vertical plane that is perpendicular to the wall.

Using the model,

A horizontal force of magnitude kmg, where k is a constant, is now applied to the beam at A. This force acts in a direction that is perpendicular to the wall and towards the wall. Given that tanθ = , μ = and the beam is now in limiting equilibrium,

use the model to find the value of k.

A rough plane is inclined to the horizontal at an angle α, where tanα = A brick P of mass m is placed on the plane. The coefficient of friction between P and the plane is μ Brick P is in equilibrium and on the point of sliding down the plane. Brick P is modelled as a particle. Using the model,

find, in terms of m and g, the magnitude of the normal reaction of the plane on brick P

For parts (c) and (d), you are not required to do any further calculations. Brick P is now removed from the plane and a much heavier brick Q is placed on the plane. The coefficient of friction between Q and the plane is also

Explain briefly why brick Q will remain at rest on the plane. Brick Q is now projected with speed 0.5 m s⁻¹ down a line of greatest slope of the plane. Brick Q is modelled as a particle. Using the model,

describe the motion of brick Q, giving a reason for your answer.

A ladder AB has mass M and length 6a. The end A of the ladder is on rough horizontal ground. The ladder rests against a fixed smooth horizontal rail at the point C. The point C is at a vertical height 4a above the ground. The vertical plane containing AB is perpendicular to the rail. The ladder is inclined to the horizontal at an angle α, where sinα = , as shown in Figure 1. The coefficient of friction between the ladder and the ground is μ. The ladder rests in limiting equilibrium. The ladder is modelled as a uniform rod. Using the model,

show that the magnitude of the force exerted on the ladder by the rail at C is

Hence, or otherwise, find the value of μ.

A particle of mass 2 kg lies on a rough plane. The plane is inclined to the horizontal at 30°. The coefficient of friction between the particle and the plane is . The particle is held in equilibrium by a force of magnitude P newtons. The force makes an angle of 20° with the horizontal and acts in a vertical plane containing a line of greatest slope of the plane, as shown in Figure 1. Find the least possible value of P.

A particle P of mass 4 m is held at rest at the point X on the surface of a rough inclined plane which is fixed to horizontal ground. The point X is a distance h from the bottom of the inclined plane. The plane is inclined to the horizontal at an angle α where tan . The coefficient of friction between P and the plane is The particle P is attached to one end of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a particle Q of mass m which hangs freely at a distance d, where d h, below the pulley, as shown in Figure 4. The string lies in a vertical plane through a line of greatest slope of the inclined plane. The system is released from rest with the string taut and P moves down the plane. For the motion of the particles before P hits the ground,

state which of the information given above implies that the magnitudes of the accelerations of the two particles are the same,

write down an equation of motion for each particle,

find the acceleration of each particle.

When P hits the ground, it immediately comes to rest. Given that Q comes to instantaneous rest before reaching the pulley,

A particle P of mass 5kg is held at rest in equilibrium on a rough inclined plane by a horizontal force of magnitude 10N. The plane is inclined to the horizontal at an angle α where tan , as shown in Figure 1. The line of action of the force lies in the vertical plane containing P and a line of greatest slope of the plane. The coefficient of friction between P and the plane is Given that P is on the point of sliding down the plane, find the value of

Two particles, A and B, have masses 2 m and m respectively. The particles are attached to the ends of a light inextensible string. Particle A is held at rest on a fixed rough horizontal table at a distance d from a small smooth light pulley which is fixed at the edge of the table at the point P. The coefficient of friction between A and the table is , where < The string is parallel to the table from A to P and passes over the pulley. Particle B hangs freely at rest vertically below P with the string taut and at a height h, (h < d), above a horizontal floor, as shown in Figure 3. Particle A is released from rest with the string taut and slides along the table.

Write down an equation of motion for

Write down an equation of motion for

Hence show that, until B hits the floor, the acceleration of A is 1 2μ).

Find, in terms of g, h and μ, the speed of A at the instant when B hits the floor.

After B hits the floor, A continues to slide along the table. Given that — and that A comes to rest at P,

find d in terms of h.

Describe what would happen if

A particle P of mass 0.4 kg is moving on rough horizontal ground when it hits a fixed vertical plane wall. Immediately before hitting the wall, P is moving with speed 4 m s^–1 in a direction perpendicular to the wall. The particle rebounds from the wall and comes to rest at a distance of 5 m from the wall. The coefficient of friction between P and the

Find the magnitude of the impulse exerted on P by the wall.

A particle P of mass 2 kg is held at rest in equilibrium on a rough plane by a constant force of magnitude 40 N. The direction of the force is inclined to the plane at an angle of 30°. The plane is inclined to the horizontal at an angle of 20°, as shown in Figure 2. The line of action of the force lies in the vertical plane containing P and a line of greatest slope of the plane. The coefficient of friction between P and the plane is Given that P is on the point of sliding up the plane, find the value of

Two particles P and Q have masses 1.5 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. Particle P is held at rest on a fixed rough horizontal table. The coefficient of friction between P and the table is . The string is parallel to the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle Q hangs freely at rest vertically below the pulley, as shown in Figure 3. Particle P is released from rest with the string taut and slides along the table. Assuming that P has not reached the pulley, find

the tension in the string during the motion,

the magnitude and direction of the resultant force exerted on the pulley by the string.

A particle P of mass 2.7 kg lies on a rough plane inclined at 40° to the horizontal. The particle is held in equilibrium by a force of magnitude 15 N acting at an angle of 50° to the plane, as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the plane. The particle is in equilibrium and is on the point of sliding down the plane. Find

the magnitude of the normal reaction of the plane on P,

the coefficient of friction between P and the plane.

The force of magnitude 15 N is removed.

Determine whether P moves, justifying your answer.

A fixed rough plane is inclined at 30° to the horizontal. A small smooth pulley P is fixed at the top of the plane. Two particles A and B, of mass 2 kg and 4 kg respectively, are attached to the ends of a light inextensible string which passes over the pulley P. The part of the string from A to P is parallel to a line of greatest slope of the plane and B hangs freely below P, as shown in Figure 2. The coefficient of friction between A and the plane is . Initially A is held at rest on the plane. The particles are released from rest with the string taut and A moves up the plane. Find the tension in the string immediately after the particles are released.

A box of mass 2 kg is held in equilibrium on a fixed rough inclined plane by a rope. The rope lies in a vertical plane containing a line of greatest slope of the inclined plane. The rope is inclined to the plane at an angle , where tan = , and the plane is at an angle of 30° to the horizontal, as shown in Figure 1. The coefficient of friction between the box and the inclined plane is and the box is on the point of slipping up the plane. By modelling the box as a particle and the rope as a light inextensible string, find the tension in the rope.

A particle P of mass 0.6 kg slides with constant acceleration down a line of greatest slope of a rough plane, which is inclined at 25° to the horizontal. The particle passes through two points A and B, where AB = 10 m, as shown in Figure 3. The speed of P at A is 2 m s^–1. The particle P takes 3.5 s to move from A to B. Find

the speed of P at B,

the acceleration of P,

the coefficient of friction between P and the plane.

A lifeboat slides down a straight ramp inclined at an angle of 15º to the horizontal. The lifeboat has mass 800 kg and the length of the ramp is 50 m. The lifeboat is released from rest at the top of the ramp and is moving with a speed of 12.6 m s^–1 when it reaches the end of the ramp. By modelling the lifeboat as a particle and the ramp as a rough inclined plane, find the coefficient of friction between the lifeboat and the ramp.

Figure 5 shows two particles A and B, of mass 2 m and 4 m respectively, connected by a light inextensible string. Initially A is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle , where tan . The coefficient of friction between A and the plane is . The string passes over a small smooth pulley P which is fixed at the top of the plane. The part of the string from A to P is parallel to a line of greatest slope of the plane and B hangs vertically below P. The system is released from rest with the string taut, with A at the point X and with B at a height h above the ground. For the motion until B hits the ground,

give a reason why the magnitudes of the accelerations of the two particles are the same,

write down an equation of motion for each particle,

find the acceleration of each particle.

Particle B does not rebound when it hits the ground and A continues moving up the plane towards P. Given that A comes to rest at the point Y, without reaching P,

find the distance XY in terms of h.

A box of mass 5 kg lies on a rough plane inclined at 30° to the horizontal. The box is held in equilibrium by a horizontal force of magnitude 20 N, as shown in Figure 2. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The box is in equilibrium and on the point of moving down the plane. The box is modelled as a particle. Find

the magnitude of the normal reaction of the plane on the box,

the coefficient of friction between the box and the plane.

A particle of weight W newtons is held in equilibrium on a rough inclined plane by a horizontal force of magnitude 4 N. The force acts in a vertical plane containing a line of

Given that the particle is on the point of sliding down the plane,

show that the magnitude of the normal reaction between the particle and the plane is 20 N,

find the value of W.

Two particles P and Q have masses 0.3 kg and m kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a fixed rough plane. The plane is inclined to the horizontal at

The string lies in a vertical plane through a line of greatest slope of the inclined plane. The particle P is held at rest on the inclined plane and the particle Q hangs freely below the pulley with the string taut, as shown in Figure 2.

Find

the magnitude of the normal reaction of the inclined plane on P,

the value of m.

When the particles have been moving for 0.5 s, the string breaks. Assuming that P does not reach the pulley,

find the further time that elapses until P comes to instantaneous rest.

A particle of weight 120 N is placed on a fixed rough plane which is inclined at an angle to the horizontal, where tan = The coefficient of friction between the particle and the plane is The particle is held at rest in equilibrium by a horizontal force of magnitude 30 N, which acts in the vertical plane containing the line of greatest slope of the plane through the particle, as shown in Figure 2.

Show that the normal reaction between the particle and the plane has magnitude 114 N.

The horizontal force is removed and replaced by a force of magnitude P newtons acting up the slope along the line of greatest slope of the plane through the particle, as shown in Figure 3. The particle remains in equilibrium.

Find the greatest possible value of P.

Find the magnitude and direction of the frictional force acting on the particle when P = 30.

Two particles A and B, of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially B is held at rest on a rough fixed plane inclined at angle to the horizontal, where tan = . The part of the string from B to P is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, P, fixed at the top of the plane. The particle A hangs freely below P, as shown in Figure 4. The coefficient of friction between B and the plane is . The particles are released from rest with the string taut and B moves up the plane.

Find the magnitude of the acceleration of B immediately after release.

Find the speed of B when it has moved 1 m up the plane.

When B has moved 1 m up the plane the string breaks. Given that in the subsequent motion B does not reach P,

find the time between the instants when the string breaks and when B comes to instantaneous rest.

A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor is . The box is pushed by a force of magnitude 100 N which acts at an angle of 30 with the floor, as shown in Figure 1. Given that the box moves with constant speed, find the mass of the box.

A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude P newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle α, where tan α = , as shown in Figure 2. The coefficient of friction between the particle and the plane is Given that the particle is on the point of sliding up the plane, find

the magnitude of the normal reaction between the particle and the plane,

the value of P.

A particle of mass 0.8 kg is held at rest on a rough plane. The plane is inclined at 30° to the horizontal. The particle is released from rest and slides down a line of greatest slope of the plane. The particle moves 2.7 m during the first 3 seconds of its motion. Find

the acceleration of the particle,

the coefficient of friction between the particle and the plane.

The particle is now held on the same rough plane by a horizontal force of magnitude X newtons, acting in a plane containing a line of greatest slope of the plane, as shown in Figure 3. The particle is in equilibrium and on the point of moving up the plane.

Find the value of X.

A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle θ, where tan , and released from rest. The coefficient of friction between the brick and the plane is Find the acceleration of the brick.

A small box of mass 15 kg rests on a rough horizontal plane. The coefficient of friction between the box and the plane is 0.2. A force of magnitude P newtons is applied to the box at 50° to the horizontal, as shown in Figure 1. The box is on the point of sliding along the plane. Find the value of P, giving your answer to 2 significant figures.

A package of mass 4 kg lies on a rough plane inclined at 30° to the horizontal. The package is held in equilibrium by a force of magnitude 45 N acting at an angle of 50° to the plane, as shown in Figure 3. The force is acting in a vertical plane through a line of greatest slope of the plane. The package is in equilibrium on the point of moving up the plane. The package is modelled as a particle. Find

the magnitude of the normal reaction of the plane on the package,

the coefficient of friction between the plane and the package.

Two particles P and Q, of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force F of magnitude 30 N is applied to Q in the direction PQ, as shown in Figure 4. The force is applied for 3 s and during this time Q travels a distance of 6 m. The coefficient of friction between each particle and the plane is . Find

the acceleration of Q,

the tension in the string.

State how in your calculation you have used the information that the string is inextensible.

When the particles have moved for 3 s, the force F is removed.

Find the time between the instant that the force is removed and the instant that Q comes to rest.

Two particles A and B, of mass m and 2 m respectively, are attached to the ends of a light inextensible string. The particle A lies on a rough horizontal table. The string passes over a small smooth pulley P fixed on the edge of the table. The particle B hangs freely below the pulley, as shown in Figure 3. The coefficient of friction between A and the table is. The particles are released from rest with the string taut. Immediately after release, the magnitude of the acceleration of A and B is . By writing down separate equations of motion for A and B,

find the tension in the string immediately after the particles begin to move,

When B has fallen a distance h, it hits the ground and does not rebound. Particle A is then

Find the speed of A as it reaches P.

State how you have used the information that the string is light.