Resolving Forces, Inclined Planes & Friction (Edexcel A Level Maths: Mechanics): Exam Questions

A wooden crate of mass 20 kg is pulled in a straight line along a rough horizontal floor using a handle attached to the crate.

The handle is inclined at an angle to the floor, as shown in Figure 1, where .

The tension in the handle is 40 N.

The coefficient of friction between the crate and the floor is 0.14.

The crate is modelled as a particle and the handle is modelled as a light rod.

Using the model, find the acceleration of the crate.

The crate is now pushed along the same floor using the handle. The handle is again inclined at the same angle to the floor, and the thrust in the handle is 40 N as shown in Figure 2 below.

Explain briefly why the acceleration of the crate would now be less than the acceleration of the crate found in part (a).

Figure 1 shows a particle of mass  suspended by a light inextensible string, with the other end of the string attached to a fixed point . 

With the string at an angle of  to the vertical, equilibrium is maintained by a horizontal force of 12 N which acts on the particle as shown in Figure 1.

Find

(i) the tension in the string,

(ii) and the value of .

A particle , of mass 7 kg, is pulled along a rough horizontal plane by a light horizontal string.

The string is inclined at 20° above the horizontal and the tension in the string is 45 N, as shown in Figure 2.

The coefficient of friction between and the plane is 0.4.

Given that the particle is moving, find the magnitude of the acceleration of to two significant figures.

A block of mass 6 kg rests on a rough horizontal surface.

A force of magnitude N acts on the block as shown in Figure 2.

The coefficient of friction between the block and the surface is 0.3.

Given that the block is on the point of sliding, calculate the magnitude of .

A particle of mass 0.70 kg rests on a rough plane that is inclined at an angle of 35° to the horizontal, as shown in Figure 1.

A force of magnitude  N acts up the line of greatest slope of the plane and keeps the particle in equilibrium, on the point of sliding up the plane.

The coefficient of friction between the particle and the plane is .

Determine the value of .

A particle of mass 3 kg rests on a fixed rough plane that is inclined at 15° to the horizontal.

The coefficient of friction between the particle and the plane is 0.2.

The particle is released from rest and slides down the line of greatest slope of the plane.

Taking , calculate the distance the particle travels down the plane in the first 3 seconds of its motion.

A particle of mass 3 kg is projected up a fixed rough plane that is inclined at 25° to the horizontal.

The particle is launched up the line of greatest slope with initial speed ms^-1.

After travelling 1.91 m up the plane it comes to instantaneous rest.

The coefficient of friction between the particle and the plane is 0.27.

Determine the value of .

Figure 1 shows a box  of mass 5 kg resting on a rough horizontal table.  It is connected by a light inextensible string to a sphere  of mass 2 kg. The string passes over a smooth light fixed pulley at the edge of the table so that  is hanging vertically downwards as shown in Figure 1.

The string between  and the pulley is horizontal, and the coefficient of friction between  and the table is 0.35.

The system is released from rest with the string taut.

As  descends, calculate

(i) the initial acceleration of the two objects

(ii) the tension in the string.

Figure 1 shows a particle of mass kg  hanging in equilibrium, suspended by two light inextensible strings. The strings are inclined at 25° and 70° to the horizontal, as shown.

Given that the tension in the string angled at 70° to the horizontal is 56 N, find the value of .

Figure 1 shows a particle of mass 12 kg being pushed up a smooth slope by a force of 50 N that acts horizontally. The slope is inclined at  to the horizontal.

Calculate the acceleration of the particle up the slope.

Calculate the normal reaction force of the slope on the particle.

A particle of mass 15 kg is sliding down a rough slope that is angled 25° to the horizontal.

The coefficient of friction between the particle and the slope is 0.3.

Calculate the acceleration of the particle down the slope.

Figure 1 shows a particle being pulled up a smooth slope by a force of 60 N that acts at an angle of  to the slope. The slope is inclined at  to the horizontal, as shown.

The particle experiences an acceleration of  up the slope.

Calculate the mass of the particle.

Calculate the normal reaction force of the slope on the particle.

Figure 1 shows two identical light cables attached symmetrically to a load of weight N.
Each cable is under the same tension N and meets the vertical at an angle .

The load is initially at rest. No forces other than its weight and the tensions in the two cables act on the system.

Write down an inequality for in terms of and that must be satisfied if the cables are to start lifting the load.

Using your result from part (a), or otherwise, show that when the cables are in a position such that , the upward acceleration of the load is still positive only if

A rough plane is inclined to the horizontal at an angle , where .

A brick of mass is placed on the plane.

The coefficient of friction between and the plane is .

Brick is in equilibrium and on the point of sliding down the plane.

Brick is modelled as a particle.

Using the model, find, in terms of and , the magnitude of the normal reaction of the plane on brick .

Show that .

For parts (c) and (d), you are not required to do any further calculations.

Brick is now removed from the plane and a much heavier brick is placed on the plane.

The coefficient of friction between and the plane is also .

Explain briefly why brick Q will remain at rest on the plane.

Brick is now projected with speed 0.5 ms⁻¹ down a line of greatest slope of the plane.

Brick is modelled as a particle.

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

A rough plane is inclined to the horizontal at an angle , where .

A small block of mass 5 kg is held in equilibrium on the plane by a horizontal force of magnitude newtons, as shown in Figure 1.

The force acts in a vertical plane which contains a line of greatest slope of the inclined plane.

The block is modelled as a particle.

The magnitude of the normal reaction of the plane on is 68.6 N.

Using the model,

(i) find the magnitude of the frictional force acting on ,

(ii) state the direction of the frictional force acting on .

The horizontal force of magnitude newtons is now removed and moves down the plane.

Given that the coefficient of friction between and the plane is 0.5,

find the acceleration of down the plane.

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

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

Initially is held at rest on a fixed rough plane.

The plane is inclined to the horizontal at an angle , where .

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

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

Stone hangs freely below , as shown in Figure 1.

The coefficient of friction between and the plane is .

Stone 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 reaches the pulley, write down an equation of motion for .

Using the model for the motion of the system before reaches the pulley, show that the acceleration of is .

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

In reality, the string is not light.

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

Figure 1 shows two particles  and , of masses 4 kg and 3 kg respectively, connected by a light inextensible string. 

Particle  is held motionless on a rough fixed plane inclined at  to the horizontal.  The string passes over a smooth light pulley fixed at the top of the plane so that  is hanging vertically downwards as shown in Figure 1.

The string between  and the pulley lies along a line of greatest slope of the plane, and  hangs freely from the pulley. 

The coefficient of friction between particle  and the plane is .

The system is released from rest with the string taut.

Calculate

(i) the initial acceleration of the two objects,

(ii) the tension in the string as  descends.

After descending for 3.2 seconds, particle  strikes the ground and immediately comes to rest.  Particle  continues to move up the slope until the forces of gravity and friction cause it to come momentarily to rest.

Find the total distance travelled by particle  between the time that the system is first released from rest and the time that particle  comes momentarily to rest again after  has struck the ground.

Figure 1 shows a particle of mass 0.9 kg on a rough horizontal plane.  A force of magnitude N is acting on the particle at an angle of 40° to the horizontal as shown.

Given that the coefficient of friction between the plane and the particle is 0.3, and that the particle is on the point of sliding to the right under the influence of the force, find the value of .

A particle of mass kg is held at rest on a rough plane inclined at angle to the horizontal, where .

The coefficient of friction between the particle and the plane is .

The particle is then released.

Given that the particle remains motionless after it is released, show that .

A particle of mass 2 kg is projected up a rough plane which is inclined at an angle of 20° to the horizontal. 

It is projected up the line of greatest slope with an initial velocity of , and it comes to instantaneous rest after moving a distance of 4.85 m up the slope. 

The coefficient of friction between the particle and the slope is 0.2.

Find the value of .

A smooth bead of mass grams is threaded on a light, inextensible string.
The ends of the string are fixed at two points and , with vertically above

The section of the string makes an angle of 35° with the vertical.

The section of the string makes an angle of 55° with the vertical.

The bead is held in equilibrium by a horizontal force of 3 N acting towards the left, as shown in Figure 1.

The only forces on the bead are its weight and the tensions in and .

Taking = 9.8 ms^-2, find the value of .

A particle of mass 0.8 kg is released from rest on a rough plane that is inclined at an angle to the horizontal, where .

After 4 s the speed of the particle is 1.35 m s^-1.

Determine the coefficient of friction, , between the particle and the plane.

Two blocks, and , of masses and respectively, are attached to the ends of a light string.

Initially is held at rest on a fixed rough plane.

The plane is included at angle to the horizontal ground, where .

The string passes over a small smooth pulley, , fixed at the top of the plane.

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

Block hangs freely below , as shown in Figure 1.

The coefficient of friction between and the plane is .

The blocks are released from rest with the string taught and moves up the plane.

The tension in the string immediately after the blocks are released is .

The blocks are modelled as particles and the string is modelled as being inextensible.

Show that .

After reaches the ground, continues to move up the plane until it comes to rest before reaching .

Determine whether will remain at rest, carefully justifying your answer.

Suggest two refinements to the model that would make it more realistic.

Two small particles and , of masses 2.7 kg and 2.2 kg respectively, are joined by a light inextensible string.

Particle is initially held at rest on a fixed rough plane that is inclined at 25° to the horizontal.

The string passes over a small smooth pulley at the top of the plane so that particle hangs freely and vertically below the pulley, as shown in Figure 1.

The section of the string parallel with the plane lies along the line of greatest slope of the plane.

The coefficient of friction between particle and the plane is .

The system is released from rest with the string taut.

Given that particle descends 1.82 m in the first 3 s after it is released, find the value of .

A small particle is placed on a rough plane that can be set at different angles to the horizontal.

The plane is first inclined at an angle where . The particle is released from rest and remains at rest on the plane.

The angle of the plane is then increased to angle where . The particle is again released from rest, and this time it begins to slide down the line of greatest slope of the plane.

The coefficient of friction between the particle and the plane is .

Show that

A particle of mass 0.5 kg rests on a fixed rough plane that is inclined at 35° to the horizontal as shown in Figure 1.

A force of 6 N acts on the particle in the same vertical plane as the line of greatest slope of the plane.

The line of action of the force makes an acute angle with the plane, as shown.

The coefficient of friction between the particle and the plane is 0.4.

The particle is on the point of sliding up the plane.

Calculate the value of .

A particle of mass  kg is projected up a rough plane which is inclined at an angle of  to the horizontal. 

It is projected up the line of greatest slope with an initial velocity of  metres per second, and it comes to instantaneous rest in  seconds after moving a distance of metres up the slope.

The coefficient of friction between the particle and the slope is .

Show that:

(i)

(ii)

After coming to instantaneous rest, the particle begins to slide back down the slope, and after  seconds it has returned to its starting point.

Find an expression for in terms of .

A group of scientists have landed on the planet Hephaestia, where the gravitational constant of acceleration has a different value than it does on Earth. 

Their spaceship contains a device which may be used to find the value of on any planet. 

In this device a particle with mass kg is connected by a light inextensible string to a light scale-pan.

A force meter with mass kg is placed in the scale-pan, and a small block with mass kg is placed on top of .

is held in place on a rough plane angled at to the horizontal.

The coefficient of friction between and the plane is .  

The string passes over a smooth light pulley fixed at the top of the plane so that the scale-pan is hanging vertically below as shown in Figure 1 below.

With the string between and the pulley lying in the line of greatest slope of the plane, is projected down the plane with a velocity of parallel to the string. 

After a time of seconds the system comes momentarily to rest, and then the scale-pan begins to descend under the force of gravity, pulling mass up the slope behind it.

When the scale-pan is initially moving upwards, the force exerted by on is denoted by .

When the scale-pan begins to descend, the force exerted by on is denoted by .

The force meter is only able to record the difference, , between these two values, where .

Use the above information to show that

For the scientists' ship’s device the following values apply.

Find the value of  on Hephaestia (the planet they are visiting), given that the value recorded for is 1.286 N. Give your answer to 4 significant figures.

Two particles and , of identical mass, are connected by means of a light inextensible string.

Particle is held motionless on a rough fixed plane inclined at 30° to the horizontal. This plane is connected at its top to another rough fixed plane which is inclined at 70° to the horizontal. 

The string passes over a smooth light pulley fixed at the top of the two planes so that is hanging downwards in contact with the second plane. This situation is shown in Figure 1.

The parts of the string between and the pulley and between and the pulley each lie along a line of greatest slope of the respective planes.

The coefficient of friction between the particles and the planes is 0.15 in both cases.

The system is released from rest with the string taut, and with particle a vertical distance of 0.75 m from the ground. 

Particle descends down the slope until it reaches the ground, at which point it immediately comes to rest.  Particle continues to move up the slope until the forces of gravity and friction cause it to come momentarily to rest.

Find the total distance travelled by particle between the time that the system is first released from rest and the time that particle comes momentarily to rest again after has reached the ground.