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AS A-level Mechanics · Edexcel 8MA0 · free A-level maths worksheet

Forces and friction

Resolving forces, equilibrium on rough and smooth inclined planes, the coefficient of friction, limiting equilibrium, motion on a slope.

12 original questions, 89 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

A particle of mass 4 kg is held at rest on a smooth plane inclined at 25° to the horizontal. It is held by a force of magnitude P newtons acting up a line of greatest slope of the plane, as shown in the diagram.

  1. Find the value of P.
  2. Find the magnitude of the normal reaction of the plane on the particle.
25°P N
4 marks
Show answer
a) P = 16.6 b) 35.5 N
2

A box of mass 6 kg rests on rough horizontal ground. A rope is attached to the box and pulled with a force of 30 N at an angle of 20° above the horizontal, as shown in the diagram. The box is on the point of moving.

The box is modelled as a particle.

  1. Find the magnitude of the normal reaction of the ground on the box.
  2. Find the coefficient of friction between the box and the ground.
20°30 N
5 marks
Show answer
a) 48.5 N b) μ = 0.581
3

A sledge of mass 12 kg is at rest on a rough slope inclined at 15° to the horizontal. The coefficient of friction between the sledge and the slope is 0.4

The sledge is modelled as a particle.

  1. Find the magnitude of the frictional force acting on the sledge.
  2. Show that the sledge is not in limiting equilibrium.
5 marks
Show answer
a) 30.4 N b) F = 30.4 N < μR = 45.4 N
4

A rough plane is inclined to the horizontal at an angle α, where tan α = 724

A small parcel of mass 3 kg is released from rest on the plane and slides down a line of greatest slope. The coefficient of friction between the parcel and the plane is 0.1

The parcel is modelled as a particle.

  1. Find the acceleration of the parcel.
  2. Find the speed of the parcel when it has slid 5 m.
α
7 marks
Show answer
a) 1.80 m s-2 b) 4.25 m s-1
5

A puck of mass 0.5 kg is projected with speed 8 m s−1 up a line of greatest slope of a rough plane inclined at 20° to the horizontal. The coefficient of friction between the puck and the plane is 0.3

The puck is modelled as a particle.

  1. Find the deceleration of the puck as it moves up the plane.
  2. Find the distance the puck travels up the plane before it comes to rest.
  3. Determine whether the puck remains at rest, giving a reason for your answer.
20°8 m s⁻¹
8 marks
Show answer
a) 6.11 m s-2 b) 5.23 m c) slides back down, since 0.5g sin 20° = 1.68 N > μR = 1.38 N
6

A block of mass 8 kg rests on a rough plane inclined at 30° to the horizontal. A horizontal force of magnitude 60 N acts on the block, in the vertical plane containing a line of greatest slope, pushing towards the plane as shown in the diagram. The block is on the point of moving up the plane.

The block is modelled as a particle.

  1. Find the magnitude of the normal reaction of the plane on the block.
  2. Find the coefficient of friction between the block and the plane.

The horizontal force is now removed.

  1. Find the acceleration of the block down the plane.
30°60 N
9 marks
Show answer
a) 97.9 N b) μ = 0.130 c) 3.79 m s-2
7

A particle of mass m is on a rough plane inclined at an angle θ to the horizontal. The coefficient of friction between the particle and the plane is μ. A force of magnitude P, acting up a line of greatest slope, pulls the particle up the plane at constant speed.

  1. Show that P = mg(sin θ + μ cos θ).

The force is replaced by a force of magnitude 12P, also acting up the line of greatest slope. The particle is now at rest and on the point of sliding down the plane.

  1. Show that μ = 13 tan θ.
7 marks
Show answer
P = mg(sin θ + μ cos θ); μ = (1/3) tan θ
8

A sledge of mass 15 kg is pulled along rough horizontal ground by a rope inclined at 30° above the horizontal. The tension in the rope is 60 N. The coefficient of friction between the sledge and the ground is 0.2

The sledge is modelled as a particle and the rope as light.

  1. Find the acceleration of the sledge.

The sledge is now pushed along the same ground by a light rod inclined at 30° below the horizontal. The thrust in the rod is 60 N.

  1. Explain why the acceleration is now less than in part (a).
  2. Find the new acceleration.
30°60 N
8 marks
Show answer
a) 1.90 m s-2 b) the push has a downward component, so R and therefore the friction μR are larger c) 1.10 m s-2
9

A small crate of mass 3 kg is on a rough ramp inclined at 35° to the horizontal. The coefficient of friction between the crate and the ramp is 0.5

A force of magnitude P newtons, acting up a line of greatest slope of the ramp, keeps the crate at rest. The crate is modelled as a particle.

  1. Explain why the crate would not stay at rest if P = 0
  2. Find the least possible value of P.
  3. Find the greatest possible value of P.
8 marks
Show answer
a) 3g sin 35° = 16.9 N > μR = 12.0 N b) P = 4.82 c) P = 28.9
10

A particle P of mass 4 kg is on a rough plane inclined at 25° to the horizontal. The coefficient of friction between P and the plane is 0.35

A force of magnitude X newtons acts on P at an angle of 30° above a line of greatest slope, in the vertical plane containing that line, as shown in the diagram. The particle is on the point of moving up the plane.

  1. Find the value of X.

The force is now removed.

  1. Show that P slides down the plane and find its acceleration.
P25°30°X N
9 marks
Show answer
a) X = 27.9 b) 4g sin 25° = 16.6 N > μR = 12.4 N so it slides; 1.03 m s-2
11

A skier of mass 70 kg passes the top of a straight slope with speed 3 m s−1, moving down a line of greatest slope. The slope is 50 m long and is inclined at 12° to the horizontal. At the bottom the skier moves onto horizontal snow, with no change of speed, and continues in a straight line until coming to rest. The coefficient of friction between the skis and the snow is 0.08 everywhere.

The skier is modelled as a particle and air resistance is ignored.

  1. Find the speed of the skier at the bottom of the slope.
  2. Find the distance the skier travels on the horizontal snow.
  3. Air resistance is now included in the model. State, with a reason, how this would change your answer to part (b).
9 marks
Show answer
a) 11.7 m s-1 b) 86.8 m c) shorter: there is an extra force opposing motion (and the skier arrives slower)
12

A rough plane is inclined to the horizontal at an angle α, where tan α = 512

A particle Q of mass m is projected with speed U up a line of greatest slope of the plane from the point A. As Q moves up the plane it decelerates at 913g. The coefficient of friction between Q and the plane is μ.

  1. Show that μ = 13
  2. Explain why Q does not remain at rest when it stops.
  3. Find, in terms of g, the acceleration of Q as it slides back down the plane.
  4. Show that Q passes back through A with speed 13U.
10 marks
Show answer
a) μ = 1/3 b) mg sin α = 5mg/13 > μR = 4mg/13 c) g/13 d) v = U/3

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Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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