Friction

The diagram shows a non-uniform rod AB of mass 4 kg and length 2 m. The end A of the rod rests against a rough vertical wall. The rod is held in a horizontal position, perpendicular to the wall, by a light inextensible string attached to the rod at B. The other end of the string is attached to the wall at a point vertically above A. The string is inclined at an angle of θ° to the horizontal, where

The rod rests in equilibrium in a vertical plane perpendicular to the wall. The rod’s weight acts at the point G on AB.

You are given that the magnitude of the moment of the rod’s weight about A is 47.04 N m.

Show that the distance AG is 1.2 m.

Find the tension in the string.

Determine the magnitude of the contact force exerted on the rod by the wall at A.

A rectangular block B of mass 10 kg lies at rest in limiting equilibrium on a rough plane Π inclined at 30° to the horizontal. A horizontal force of magnitude T N, acting above a line of greatest slope, is applied to B (see Fig. 1).

The coefficient of friction between B and the plane is 0.8.

Show that the value of T is 14.9, correct to 3 significant figures.

For the remainder of the question, you should you use this value of T.

Block B is now cut at an angle of 30° to the horizontal into two smaller blocks. The upper block has a mass of 4 kg, and the lower block has a mass of 6 kg. The two blocks are held at rest with the lower block on Π. The horizontal force of magnitude T N is now applied to the lower block (see Fig. 2).

The two blocks are released from rest and in the subsequent motion the upper block starts to move with acceleration 3.5 m s^-2.

Determine the coefficient of friction between the two blocks.

Show that the lower block does not move.

By taking moments, show that the magnitude of the normal contact force between the lamina

and the wall is 1.24 N, correct to 3 significant figures.

Determine the coefficient of friction between the lamina and the ground.

The points A and B are the lower and upper ends, respectively, of a line of greatest slope on a plane inclined at an angle θ to the horizontal, where sin θ = 0.6 and AB = 1.375 m (see diagram).

A particle P is projected up the plane with speed 6 ms^-1 from A towards B.

The plane at A is fixed to the ground which is horizontal.

The surface of the plane is rough and the coefficient of friction between P and the plane is 0.25.

Show that the speed of P at B is 3.8 ms^-1.

The particle leaves the slope at B and moves freely under gravity.

The particle first lands at a point C on the horizontal ground. The time taken for P to travel from A to C is T seconds.

Determine the value of T.

A child attempts to drag a sledge along horizontal ground by means of a rope attached to the sledge. The rope makes an angle of 15° with the horizontal (see diagram).

Given that the sledge remains at rest and that the frictional force acting on the sledge is 60 N, find

the tension in the rope.

A rectangular block B is at rest on a horizontal surface. A particle P of mass 2.5 kg is placed on the upper surface of B. The particle P is attached to one end of a light inextensible string which passes over a smooth fixed pulley. A particle Q of mass 3 kg is attached to the other end of the string and hangs freely below the pulley. The part of the string between P and the pulley is horizontal (see diagram).

The particles are released from rest with the string taut. It is given that B remains in equilibrium while P moves on the upper surface of B. The tension in the string while P moves on B is 16.8 N.

Find the acceleration of Q while P and B are in contact.

Determine the coefficient of friction between P and B.

A uniform rod AB of mass 4 kg and length 3 m rests in a vertical plane with A on rough horizontal ground.

A particle of mass 1 kg is attached to the rod at B. The rod makes an angle of 60° with the horizontal and is held in limiting equilibrium by a light inextensible string CD. D is a fixed point vertically above A and CD makes an angle of 60° with the vertical. The distance AC is x m (see diagram).

Find, in terms of g and x, the tension in the string.

Determine the value of x.

A block D of weight 50 N lies at rest in equilibrium on a fixed rough horizontal surface. A force of magnitude 15 N is applied to D at an angle θ to the horizontal (see diagram).

Complete the diagram in the Printed Answer Booklet showing all the forces acting on D.

It is given that D remains at rest and the coefficient of friction between D and the surface is 0.2.

Show that

15 cos θ - 3 sin θ ⩽ 10.

One end of a light inextensible string is attached to a particle A of mass 2 kg. The other end of the string is attached to a second particle B of mass 3 kg. Particle A is in contact with a smooth plane inclined at 30° to the horizontal and particle B is in contact with a rough horizontal plane.

A second light inextensible string is attached to B. The other end of this second string is attached to a third particle C of mass 4 kg. Particle C is in contact with a smooth plane Π inclined at an angle of 60° to the horizontal.

Both strings are taut and pass over small smooth pulleys that are at the tops of the inclined planes. The parts of the strings from A to the pulley, and from C to the pulley, are parallel to lines of greatest slope of the corresponding planes (see diagram).

The coefficient of friction between B and the horizontal plane is μ. The system is released from rest and in the subsequent motion C moves down Π with acceleration a m s^-2.

By considering an equation involving μ, a and g show that a < (2 - 1).

Given that a = , determine the magnitude of the contact force between B and the horizontal

plane. Give your answer correct to 3 significant figures.

One end of a light inextensible string is attached to a particle A of mass 2 kg. The other end of the string is attached to a second particle B of mass 2.5 kg. Particle A is in contact with a rough plane inclined at θ to the horizontal, where cos θ = . The string is taut and passes over a small smooth pulley P 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. Particle B hangs freely below P at a distance 1.5 m above horizontal ground, as shown in the diagram.

The coefficient of friction between A and the plane is μ. The system is released from rest and in the subsequent motion B hits the ground before A reaches P. The speed of B at the instant that it hits the ground is 1.2 ms^-1.

For the motion before B hits the ground, show that the acceleration of B is 0.48 ms^-2.

For the motion before B hits the ground, show that the tension in the string is 23.3 N.

Determine the value of μ.

After B hits the ground, A continues to travel up the plane before coming to instantaneous rest before it reaches P.

Determine the distance that A travels from the instant that B hits the ground until A comes to

instantaneous rest.

The diagram shows a small block B, of mass 0.2 kg, and a particle P, of mass 0.5 kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane.

The block can move on the horizontal surface, which is rough. The particle can move on the inclined plane, which is smooth and which makes an angle of θ with the horizontal where tan θ =

The system is released from rest. In the first 0.4 seconds of the motion P moves 0.3 m down the plane and B does not reach the pulley.

Find the tension in the string during the first 0.4 seconds of the motion.

Calculate the coefficient of friction between B and the horizontal surface.

The diagram shows a ladder AB, of length 2 a and mass m, resting in equilibrium on a vertical wall of height h. The ladder is inclined at an angle of 30° to the horizontal. The end A is in contact with horizontal ground. An object of mass 2 m is placed on the ladder at a point C where AC = d.

The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth.

Hence find, in terms of a, the greatest possible value of d.

State one improvement that could be made to the model.

One end of a light inextensible string is attached to a particle A of mass m kg. The other end of the string is attached to a second particle B of mass λ m kg, where λ is a constant. Particle A is in contact with a rough plane inclined at 30° to the horizontal. The string is taut and passes over a small smooth pulley P 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. The particle B hangs freely below P (see diagram).

The coefficient of friction between A and the plane is μ.

It is given that A is on the point of moving down the plane.

Find the exact value of μ when λ =