Browse Forces & Newton's laws

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Force diagrams, resultant force & equilibrium

12 questions

LessonNot started
Two perpendicular forces of 55 N and 1212 N act on a particle. Find the magnitude of the resultant.
●●●●●Level 32 marksStart
Two forces, 66 N due east and 88 N due north, act on a particle. Find the magnitude of the resultant.
●●●●●Level 32 marksStart
A woman of mass 6060 kg stands on the floor of a lift. The lift has mass 400400 kg and is raised by a vertical cable.

Take g=9.8g = 9.8 m s⁻².
(a) Find the tension in the cable when the lift is accelerating upwards at 1.21.2 m s⁻². (3 marks)
(b) Find the magnitude of the force exerted by the floor of the lift on the woman at that instant. (3 marks)
(c) Later, while still moving upwards, the lift decelerates at 1.51.5 m s⁻². Find the magnitude of the force exerted by the floor on the woman now. (2 marks)
●●●●●Level 38 marksStart
A particle of mass 44 kg experiences a resultant force of 1212 N. Find its acceleration.
●●●●●Level 32 marksStart
θ15 NABCFigure 1 (not accurately drawn)

A small smooth ring RR is threaded on a light inextensible string. The ends of the string are attached to two fixed points AA and BB on a horizontal ceiling.

A horizontal force of magnitude 1515 N acts on RR. The ring rests in equilibrium at the point CC, vertically below BB, with the string taut, as shown in Figure 1. The horizontal force and the string lie in the same vertical plane.

The part ACAC of the string makes an angle θ\theta with the vertical, where tan⁡θ=512\tan\theta = \frac{5}{12}

The tension in the string is TT newtons and the mass of RR is MM kg.
(a) Find the value of TT

(3 marks)
(b) Find the value of MM

(3 marks)
●●●●●Level 36 marksStart

Newton's laws and F = ma

43 questions

LessonNot started
A car of mass 10001000 kg decelerates at 3 m s−23\ \text{m s}^{-2}. Find the magnitude of the braking force.
●●●●●Level 32 marksStart
A woman of mass 6060 kg stands on the floor of a lift. The lift has mass 400400 kg and is raised by a vertical cable.

Take g=9.8g = 9.8 m s⁻².
(a) Find the tension in the cable when the lift is accelerating upwards at 1.21.2 m s⁻². (3 marks)
(b) Find the magnitude of the force exerted by the floor of the lift on the woman at that instant. (3 marks)
(c) Later, while still moving upwards, the lift decelerates at 1.51.5 m s⁻². Find the magnitude of the force exerted by the floor on the woman now. (2 marks)
●●●●●Level 38 marksStart
A box of mass 88 kg rests on a horizontal floor. Find the normal reaction from the floor. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
Figure 1ABDiagram not accurately drawn.

A lift is raised and lowered by a vertical cable which is attached to the top of the lift. The lift has mass MM.

A crate AA of mass 3m3m rests on the floor of the lift and a crate BB of mass 2m2m rests on top of crate AA, as shown in Figure 1.

The cable is modelled as being light and inextensible, air resistance is modelled as being negligible and the crates are modelled as particles.

The lift is moving vertically downwards and is slowing down with deceleration of magnitude g4\frac{g}{4}. The tension in the cable is 20mg20mg.

Using the model,
(a) find MM in terms of mm,

(4 marks)
(b) find, in terms of mm and gg, the magnitude of the force exerted on crate AA by crate BB.

(3 marks)
●●●●●Level 37 marksStart
A car of mass 12001200 kg tows a trailer of mass 300300 kg along a straight horizontal road, connected by a light rigid tow-bar. The car's engine produces a driving force of 27002700 N. The resistances to motion are 400400 N on the car and 200200 N on the trailer, and these are modelled as constant.
(a) Find the acceleration of the car and trailer. (3 marks)
(b) Find the tension in the tow-bar. (3 marks)
(c) When the car and trailer are travelling at 2020 m s⁻¹, the tow-bar breaks. Assuming the resistance on the trailer is unchanged, find the distance the trailer travels after the tow-bar breaks before coming to rest. (2 marks)
●●●●●Level 38 marksStart
A particle moves under the action of two forces, F1\mathbf{F}_1 and F2\mathbf{F}_2, only.

It is given that

F1=(6i−9j) NF2=(ki+4kj) N\mathbf{F}_1 = (6\mathbf{i} - 9\mathbf{j})\ \text{N} \qquad \mathbf{F}_2 = (k\mathbf{i} + 4k\mathbf{j})\ \text{N}

where kk is a constant.

The acceleration of the particle is (2i+3j) m s−2(2\mathbf{i} + 3\mathbf{j})\ \text{m s}^{-2}

Find kk

(4 marks)
●●●●●Level 34 marksStart
A particle, of mass 44 kilograms, moves under the action of two forces, F1\mathbf{F}_1 and F2\mathbf{F}_2, only.

It is given that

F1=[a6]newtonsF2=[5b]newtons\mathbf{F}_1 = \begin{bmatrix} a \\ 6 \end{bmatrix} \text{newtons} \qquad \mathbf{F}_2 = \begin{bmatrix} 5 \\ b \end{bmatrix} \text{newtons}

where aa and bb are constants.

The acceleration of the particle is [−ba−2] m s−2\begin{bmatrix} -b \\ a - 2 \end{bmatrix}\ \text{m s}^{-2}

Find the value of aa and the value of bb

(4 marks)
●●●●●Level 34 marksStart
A parcel of mass 1.51.5 kg is released from rest from a drone that is hovering at a height of 4040 m above horizontal ground.

In a first model, the parcel is modelled as a particle moving freely under gravity.
(a) Using this first model, find the time taken for the parcel to reach the ground.

(3 marks)

In a second model, the parcel is modelled as a particle falling against a constant resistance to motion of magnitude RR newtons. Using this second model, the parcel takes 3.53.5 s to reach the ground.
(b) Find the value of RR.

(5 marks)
●●●●●Level 38 marksStart
A motorcycle and its rider have a combined mass of 250250 kg. The motorcycle is moving down a straight road which is inclined at an angle α\alpha to the horizontal, where sin⁡α=125\sin\alpha = \frac{1}{25}

The engine of the motorcycle is working at a constant rate of 66 kW. The resistance to the motion of the motorcycle from non-gravitational forces is modelled as a constant force of magnitude 420420 N.

The motorcycle and its rider are modelled as a particle.

Find the acceleration of the motorcycle at the instant when its speed is 12 m s−112\ \text{m s}^{-1}

(5 marks)
●●●●●Level 35 marksStart
[In this question i\mathbf{i} and j\mathbf{j} are horizontal unit vectors.]

A sledge of mass 22 kg is pulled across horizontal ice by three horizontal ropes. The sledge is modelled as a particle and the ice is modelled as smooth. The forces exerted on the sledge by the ropes are F1\mathbf{F}_1, F2\mathbf{F}_2 and F3\mathbf{F}_3, where

F1=(4i−7j) NF2=(−9i+3j) NF3=(2i+8j) N\mathbf{F}_1 = (4\mathbf{i} - 7\mathbf{j})\text{ N} \qquad \mathbf{F}_2 = (-9\mathbf{i} + 3\mathbf{j})\text{ N} \qquad \mathbf{F}_3 = (2\mathbf{i} + 8\mathbf{j})\text{ N}
(a) Find the magnitude of the acceleration of the sledge.

(4 marks)

A fourth horizontal force, F4=(2pi+pj)\mathbf{F}_4 = (2p\mathbf{i} + p\mathbf{j}) N, where pp is a constant, now also acts on the sledge.

The resultant of the four forces acts in the direction of the vector (5i+8j)(5\mathbf{i} + 8\mathbf{j}).
(b) Find the value of pp.

(4 marks)
●●●●●Level 38 marksStart
A person of mass 7070 kg stands in a lift that accelerates upwards at 2 m s−22\ \text{m s}^{-2}. Find the normal reaction from the floor on the person. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
A particle of mass 44 kg is pulled by a forward force of 2020 N against a resistance of 88 N. Find its acceleration.
●●●●●Level 32 marksStart
20°α°600 N450 Ndirection of motion →Figure 1 (not accurately drawn)

A yacht is towed along the centre of a long straight harbour channel by two motor launches. Each launch is attached to the same point on the yacht by a rope.
  • The tension in the first rope is 600600 N and the rope makes an angle of 20°20° with the direction of motion of the yacht
  • The tension in the second rope is 450450 N and the rope makes an angle of α°\alpha° with the direction of motion of the yacht, on the other side of the line of motion, as shown in Figure 1
Both ropes are horizontal and the yacht moves in a straight line along the centre of the channel. The yacht is modelled as a particle.
(a) Find the value of α\alpha

(3 marks)

The mass of the yacht is 2.52.5 tonnes and the resistance to the motion of the yacht is a constant force of magnitude 580580 N.
(b) Find the acceleration of the yacht.

(4 marks)
●●●●●Level 37 marksStart

Connected particles

19 questions

LessonNot started
Two particles AA (33 kg) and BB (22 kg) are joined by a light inextensible string on a smooth horizontal table. A horizontal force of 1515 N is applied to BB, away from AA.
(a) Find the acceleration of the system.
(b) Find the tension in the string.
●●●●●Level 34 marksStart
A car of mass 10001000 kg tows a trailer of mass 500500 kg along a straight level road. The engine provides a driving force of 45004500 N and there is no resistance.
(a) Find the acceleration.
(b) Find the tension in the tow bar.
●●●●●Level 34 marksStart
Two particles of mass 22 kg and 33 kg are connected by a light string on a smooth table and pulled by a force FF applied to the 33 kg particle. The system accelerates at 4 m s−24\ \text{m s}^{-2}. Find FF and the tension.
●●●●●Level 34 marksStart
Figure 1ABDiagram not accurately drawn.

A lift is raised and lowered by a vertical cable which is attached to the top of the lift. The lift has mass MM.

A crate AA of mass 3m3m rests on the floor of the lift and a crate BB of mass 2m2m rests on top of crate AA, as shown in Figure 1.

The cable is modelled as being light and inextensible, air resistance is modelled as being negligible and the crates are modelled as particles.

The lift is moving vertically downwards and is slowing down with deceleration of magnitude g4\frac{g}{4}. The tension in the cable is 20mg20mg.

Using the model,
(a) find MM in terms of mm,

(4 marks)
(b) find, in terms of mm and gg, the magnitude of the force exerted on crate AA by crate BB.

(3 marks)
●●●●●Level 37 marksStart
A car of mass 12001200 kg tows a trailer of mass 300300 kg along a straight horizontal road, connected by a light rigid tow-bar. The car's engine produces a driving force of 27002700 N. The resistances to motion are 400400 N on the car and 200200 N on the trailer, and these are modelled as constant.
(a) Find the acceleration of the car and trailer. (3 marks)
(b) Find the tension in the tow-bar. (3 marks)
(c) When the car and trailer are travelling at 2020 m s⁻¹, the tow-bar breaks. Assuming the resistance on the trailer is unchanged, find the distance the trailer travels after the tow-bar breaks before coming to rest. (2 marks)
●●●●●Level 38 marksStart
A car of mass 12001200 kg tows a trailer of mass 400400 kg. The driving force is 40004000 N, and resistances of 600600 N (car) and 200200 N (trailer) act. Find
(a) the acceleration and (b) the tension in the tow bar.
●●●●●Level 34 marksStart
A tractor tows a log of mass 300300 kg by a rope. The log accelerates at 0.5 m s−20.5\ \text{m s}^{-2} against a ground resistance of 250250 N. Find the tension in the rope.
●●●●●Level 32 marksStart
A locomotive of mass 60006000 kg pulls a single carriage of mass 40004000 kg. The driving force is 2500025000 N and resistances are negligible.
(a) Find the acceleration.
(b) Find the force in the coupling.
●●●●●Level 34 marksStart

Pulleys

22 questions

LessonNot started
Two particles of mass 77 kg and mm kg hang over a smooth pulley. The system accelerates at 1.4 m s−21.4\ \text{m s}^{-2}, with the 77 kg mass descending. Find mm. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
Particles of mass 88 kg and 22 kg hang over a smooth pulley, released from rest. Find the acceleration. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
Particles of mass 66 kg and 44 kg hang over a smooth pulley and are released from rest. Find
(a) the acceleration and (b) the tension. (g=9.8g = 9.8.)
●●●●●Level 34 marksStart
Particles of mass 44 kg and 66 kg hang over a smooth pulley. Find the force exerted by the string on the pulley while the system moves. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
Particles of mass 55 kg and 33 kg hang from the ends of a light inextensible string over a smooth fixed pulley, and are released from rest.
(a) Find the acceleration of the system.
(b) Find the tension in the string. (g=9.8g = 9.8.)
●●●●●Level 34 marksStart
Particles of mass 55 kg and 33 kg hang over a smooth pulley and are released from rest. After the 55 kg mass has descended, find the distance it falls in the first 22 seconds (assuming it does not reach the pulley or floor). (g=9.8g = 9.8.)
●●●●●Level 32 marksStart

Resolving forces

42 questions

LessonNot started
Forces of 88 N due east and 66 N due north act on a particle. Find the magnitude and bearing (from east) of the resultant.
●●●●●Level 33 marksStart
A force of 3030 N acts on a box at 20∘20^\circ above the horizontal. Find the component of the force in the horizontal direction, to 3 s.f.
●●●●●Level 32 marksStart
Two forces of 1212 N and 88 N act on a particle with an angle of 60∘60^\circ between them. Find the magnitude of the resultant, to 3 s.f.
●●●●●Level 32 marksStart
2.4 mθABDFigure 1

A uniform rod ABAB, of length 2.42.4 m and weight 5050 N, has its end AA freely hinged to a fixed point on horizontal ground. The rod is held in equilibrium at an angle θ\theta to the ground, where tan⁡θ=52\tan\theta = \frac{5}{2}, by a light inextensible string. One end of the string is attached to the rod at BB and the other end is attached to the point DD on a vertical wall. The string is horizontal, and the rod and the string lie in a vertical plane perpendicular to the wall, as shown in Figure 1.
(a) Find the tension in the string.

(3 marks)
(b) Find the magnitude of the force exerted on the rod by the hinge.

(2 marks)
(c) Find the angle between this force and the ground.

(2 marks)
●●●●●Level 37 marksStart
θABFigure 1

A uniform ladder ABAB, of weight WW and length 2a2a, rests with its end AA on rough horizontal ground and its end BB against a smooth vertical wall. The ladder lies in a vertical plane perpendicular to the wall and makes an angle θ\theta with the ground, as shown in Figure 1. The ladder rests in equilibrium.

The resultant force exerted on the ladder by the ground acts at an angle β\beta to the horizontal.
(a) Show that tan⁡β=2tan⁡θ\tan\beta = 2\tan\theta

(5 marks)
(b) Given that β=80∘\beta = 80^\circ, find the value of θ\theta.

(2 marks)
●●●●●Level 37 marksStart
A particle of weight 5050 N hangs in equilibrium, held by a horizontal force PP and a light string at 60∘60^\circ to the horizontal with tension TT. Find TT and PP, to 3 s.f.
●●●●●Level 34 marksStart
A mass of 1010 kg hangs in equilibrium, held by a light string at 50∘50^\circ to the vertical and a horizontal force PP. Taking g=9.8g = 9.8, find the tension TT and the force PP, to 3 s.f.
●●●●●Level 34 marksStart
A particle of weight 100100 N hangs in equilibrium from a light string making angle θ\theta with the vertical, held aside by a horizontal force. The tension in the string is 200200 N.
(a) Find the angle θ\theta.
(b) Find the horizontal force, to 3 s.f.
●●●●●Level 34 marksStart
A particle of weight 5050 N hangs in equilibrium, supported by two light strings. One string makes an angle of 30∘30^\circ with the horizontal and has tension T1T_1 newtons; the other makes an angle of 60∘60^\circ with the horizontal and has tension T2T_2 newtons.
(a) By resolving horizontally, show that T2=3 T1T_2 = \sqrt{3}\,T_1.
(b) Find the values of T1T_1 and T2T_2, giving each answer correct to 3 significant figures.
●●●●●Level 36 marksStart
[In this question i\mathbf{i} and j\mathbf{j} are horizontal unit vectors.]

A sledge of mass 22 kg is pulled across horizontal ice by three horizontal ropes. The sledge is modelled as a particle and the ice is modelled as smooth. The forces exerted on the sledge by the ropes are F1\mathbf{F}_1, F2\mathbf{F}_2 and F3\mathbf{F}_3, where

F1=(4i−7j) NF2=(−9i+3j) NF3=(2i+8j) N\mathbf{F}_1 = (4\mathbf{i} - 7\mathbf{j})\text{ N} \qquad \mathbf{F}_2 = (-9\mathbf{i} + 3\mathbf{j})\text{ N} \qquad \mathbf{F}_3 = (2\mathbf{i} + 8\mathbf{j})\text{ N}
(a) Find the magnitude of the acceleration of the sledge.

(4 marks)

A fourth horizontal force, F4=(2pi+pj)\mathbf{F}_4 = (2p\mathbf{i} + p\mathbf{j}) N, where pp is a constant, now also acts on the sledge.

The resultant of the four forces acts in the direction of the vector (5i+8j)(5\mathbf{i} + 8\mathbf{j}).
(b) Find the value of pp.

(4 marks)
●●●●●Level 38 marksStart
20°α°600 N450 Ndirection of motion →Figure 1 (not accurately drawn)

A yacht is towed along the centre of a long straight harbour channel by two motor launches. Each launch is attached to the same point on the yacht by a rope.
  • The tension in the first rope is 600600 N and the rope makes an angle of 20°20° with the direction of motion of the yacht
  • The tension in the second rope is 450450 N and the rope makes an angle of α°\alpha° with the direction of motion of the yacht, on the other side of the line of motion, as shown in Figure 1
Both ropes are horizontal and the yacht moves in a straight line along the centre of the channel. The yacht is modelled as a particle.
(a) Find the value of α\alpha

(3 marks)

The mass of the yacht is 2.52.5 tonnes and the resistance to the motion of the yacht is a constant force of magnitude 580580 N.
(b) Find the acceleration of the yacht.

(4 marks)
●●●●●Level 37 marksStart

Forces on an inclined plane

29 questions

LessonNot started
A particle slides down a smooth slope with acceleration 4.9 m s−24.9\ \text{m s}^{-2}. Find the angle of the slope. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
A particle is released from rest on a smooth slope at 40∘40^\circ. Find the distance it travels in the first 22 seconds, to 3 s.f. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
A particle of mass 22 kg is pulled up a smooth slope of 25∘25^\circ by a force of 1515 N acting up the slope. Find its acceleration, to 3 s.f. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
A particle of mass 44 kg is held at rest on a smooth slope of 30∘30^\circ by a force acting up the slope. Find the magnitude of this force. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
A particle of mass 33 kg is held in equilibrium on a smooth plane inclined at 30∘30^\circ to the horizontal by a horizontal force of magnitude PP newtons, as shown in the diagram. (g=9.8 m s−2g = 9.8\ \text{m s}^{-2}.)3 kg30°P(a) Show that P=17.0P = 17.0 correct to 3 significant figures. (4 marks)
(b) Find the magnitude of the normal reaction of the plane on the particle. (3 marks)
(c) The force is suddenly removed. Find the magnitude of the acceleration of the particle immediately afterwards, and state its direction. (2 marks)
●●●●●Level 39 marksStart
A particle is projected up a smooth slope of 30∘30^\circ with speed 5 m s−15\ \text{m s}^{-1}. Find the distance it travels up the slope before coming to rest, to 3 s.f. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
A rough ramp is fixed to horizontal ground. The ramp is inclined to the horizontal at an angle α\alpha, where sin⁡α=513\sin\alpha = \frac{5}{13}

A sack of sand of mass 1212 kg is dragged at constant speed through a distance of 7.57.5 m up a line of greatest slope of the ramp. The sack is dragged by a rope which is parallel to the line of greatest slope.

The coefficient of friction between the sack and the ramp is 14\frac{1}{4}

The sack is modelled as a particle and the rope is modelled as being light.

Use the model to find the total work done by the rope in dragging the sack the 7.57.5 m up the ramp.

(5 marks)
●●●●●Level 35 marksStart
A motorcycle and its rider have a combined mass of 250250 kg. The motorcycle is moving down a straight road which is inclined at an angle α\alpha to the horizontal, where sin⁡α=125\sin\alpha = \frac{1}{25}

The engine of the motorcycle is working at a constant rate of 66 kW. The resistance to the motion of the motorcycle from non-gravitational forces is modelled as a constant force of magnitude 420420 N.

The motorcycle and its rider are modelled as a particle.

Find the acceleration of the motorcycle at the instant when its speed is 12 m s−112\ \text{m s}^{-1}

(5 marks)
●●●●●Level 35 marksStart
A box of mass 1010 kg is pulled up a smooth slope of 35∘35^\circ by a force PP acting up the slope, accelerating at 2 m s−22\ \text{m s}^{-2}.
(a) Find the normal reaction.
(b) Find PP. Give answers to 3 s.f. (g=9.8g = 9.8.)mmgRP
●●●●●Level 34 marksStart

Friction

68 questions

LessonNot started
3.5 m1.5 m65°ABCFigure 1

A uniform ladder ABAB, of length 55 m and mass 2020 kg, rests with its end AA on rough horizontal ground and its end BB against a smooth vertical wall. The ladder is inclined at 65∘65^\circ to the ground and lies in a vertical plane perpendicular to the wall. A woman of mass 6060 kg stands on the ladder at the point CC, where AC=3.5AC = 3.5 m, as shown in Figure 1.

The ladder is modelled as a uniform rod and the woman as a particle.

Given that the ladder rests in limiting equilibrium, find the coefficient of friction between the ladder and the ground.

(6 marks)
●●●●●Level 36 marksStart
3 m20°ABFigure 1

A uniform rod ABAB, of length 33 m and weight 9090 N, rests with its end AA on rough horizontal ground and its end BB against a smooth vertical wall. The rod lies in a vertical plane perpendicular to the wall and is inclined to the wall at an angle of 20∘20^\circ, as shown in Figure 1. The rod rests in equilibrium.
(a) Find the magnitude of the normal reaction of the wall on the rod.

(3 marks)

The coefficient of friction between the rod and the ground is μ\mu.
(b) Find the smallest possible value of μ\mu.

(2 marks)
●●●●●Level 35 marksStart
60°ABFigure 1

A uniform ladder ABAB, of length 44 m and mass 1515 kg, rests with its end AA on rough horizontal ground and its end BB against a smooth vertical wall. The ladder is inclined at 60∘60^\circ to the ground and lies in a vertical plane perpendicular to the wall, as shown in Figure 1. The coefficient of friction between the ladder and the ground is 12\frac{1}{2}
(a) Find the magnitude of the normal reaction of the wall on the ladder.

(3 marks)

A horizontal force of magnitude PP newtons is now applied to the ladder at AA, towards the wall. The line of action of this force is perpendicular to the wall and the ladder remains in equilibrium.
(b) Find the greatest possible value of PP.

(3 marks)
●●●●●Level 36 marksStart
4 m1 mθABCFigure 1

A uniform rod ABAB, of length 55 m and weight 8080 N, rests with its end AA on rough horizontal ground. The rod rests on a small smooth horizontal peg at CC, where AC=4AC = 4 m. A particle of weight 2020 N is attached to the rod at BB. The rod lies in a vertical plane perpendicular to the peg and is inclined at an angle θ\theta to the ground, where tan⁡θ=34\tan\theta = \frac34, as shown in Figure 1. The rod is in equilibrium.
(a) Show that the magnitude of the force exerted on the rod by the peg is 6060 N.

(3 marks)

The coefficient of friction between the rod and the ground is μ\mu.
(b) Find the smallest possible value of μ\mu.

(4 marks)
●●●●●Level 37 marksStart
A block of mass 44 kg on a rough horizontal surface (μ=0.25\mu = 0.25) is pushed by a horizontal force of 3030 N and slides.

Find the acceleration of the block. (4 marks)
●●●●●Level 34 marksStart
A particle is on the point of sliding down a rough slope inclined at 20∘20^\circ. Find the coefficient of friction, to 3 s.f.
●●●●●Level 32 marksStart
A crate of mass 33 kg is dragged across a rough horizontal floor (μ=0.6\mu = 0.6) at constant speed by a horizontal rope.

Find the tension in the rope. (3 marks)
●●●●●Level 33 marksStart
The block of mass 44 kg (μ=0.3\mu = 0.3) on a rough horizontal surface is now pushed by a horizontal force of 2020 N. Find its acceleration, to 3 s.f. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
A rough ramp is fixed to horizontal ground. The ramp is inclined to the horizontal at an angle α\alpha, where sin⁡α=513\sin\alpha = \frac{5}{13}

A sack of sand of mass 1212 kg is dragged at constant speed through a distance of 7.57.5 m up a line of greatest slope of the ramp. The sack is dragged by a rope which is parallel to the line of greatest slope.

The coefficient of friction between the sack and the ramp is 14\frac{1}{4}

The sack is modelled as a particle and the rope is modelled as being light.

Use the model to find the total work done by the rope in dragging the sack the 7.57.5 m up the ramp.

(5 marks)
●●●●●Level 35 marksStart

Statics — equilibrium under several forces

31 questions

LessonNot started
A particle of mass 22 kg rests on a rough slope of 20∘20^\circ with coefficient of friction 0.30.3. Find the least force, applied up the slope, needed to stop it sliding down, to 3 s.f. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
A particle of mass 44 kg rests on a rough slope inclined at 25∘25^\circ and is on the point of sliding down. Find the coefficient of friction, to 3 s.f.
●●●●●Level 32 marksStart
A box of mass 1010 kg rests on a horizontal floor. A force of 3030 N is applied at 40∘40^\circ below the horizontal (pushing down into the floor). Find the normal reaction, to 3 s.f. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
A particle of weight 6060 N rests on a rough horizontal plane with coefficient of friction 0.250.25. It is pulled by a light string at 20∘20^\circ above the horizontal and is on the point of moving. Find the tension, to 3 s.f.
●●●●●Level 33 marksStart
A particle of weight 4040 N hangs in equilibrium from two light strings making 30∘30^\circ and 60∘60^\circ with the horizontal, on opposite sides. Find the tensions T1T_1 (in the 30∘30^\circ string) and T2T_2 (in the 60∘60^\circ string), to 3 s.f.
●●●●●Level 33 marksStart
A particle is on the point of sliding down a rough slope. The coefficient of friction is 0.50.5. Find the angle of the slope, to 3 s.f.
●●●●●Level 32 marksStart
A particle of weight 3030 N rests in equilibrium on a rough horizontal plane. A horizontal force of 1212 N acts on the particle.

Find the magnitude of the frictional force and of the normal reaction. (3 marks)
●●●●●Level 33 marksStart
A box of mass 66 kg rests on a rough horizontal floor. A horizontal force of 2020 N is just sufficient to make the box begin to slide.

Find the coefficient of friction between the box and the floor. (3 marks)
●●●●●Level 33 marksStart
A particle of weight 2020 N hangs in equilibrium from two light strings. One string makes an angle of 30∘30^{\circ} with the horizontal and the other makes 60∘60^{\circ} with the horizontal.

Find the tension in each string, giving exact answers. (6 marks)
●●●●●Level 36 marksStart
A box of mass 55 kg rests on a rough horizontal floor with coefficient of friction 0.40.4. A force PP is applied at 30∘30^{\circ} above the horizontal, and the box is on the point of sliding.

Find PP, giving your answer to 3 significant figures. (6 marks)
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A particle of weight 5050 N rests on a rough horizontal plane. A force of 2020 N is applied at 30∘30^\circ above the horizontal and the particle stays at rest. Find
(a) the normal reaction and (b) the friction force, to 3 s.f. (g=9.8g = 9.8.)
●●●●●Level 34 marksStart
Figure 1 shows a uniform rod ABAB of mass MM and length 2a2a.

The end AA of the rod rests on rough horizontal ground and the end BB rests against a smooth vertical wall.

A particle of mass 2M2M is attached to the rod at the point CC, where AC=1.5aAC = 1.5a

The rod rests in equilibrium in a vertical plane perpendicular to the wall, at an angle θ\theta to the horizontal ground, as shown in Figure 1.1.5aθABCDiagram not accurately drawn.

The magnitude of the normal reaction of the wall on the rod at BB is SS.

By taking moments about AA, show that

S=2Mgcot⁡θS = 2Mg\cot\theta

(3 marks)
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θ15 NABCFigure 1 (not accurately drawn)

A small smooth ring RR is threaded on a light inextensible string. The ends of the string are attached to two fixed points AA and BB on a horizontal ceiling.

A horizontal force of magnitude 1515 N acts on RR. The ring rests in equilibrium at the point CC, vertically below BB, with the string taut, as shown in Figure 1. The horizontal force and the string lie in the same vertical plane.

The part ACAC of the string makes an angle θ\theta with the vertical, where tan⁡θ=512\tan\theta = \frac{5}{12}

The tension in the string is TT newtons and the mass of RR is MM kg.
(a) Find the value of TT

(3 marks)
(b) Find the value of MM

(3 marks)
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A particle rests on a rough plane inclined at 35∘35^{\circ} to the horizontal, and is on the point of sliding down.

Find the coefficient of friction between the particle and the plane. (3 marks)
●●●●●Level 33 marksStart

Connected particles on slopes and pulleys

19 questions

LessonNot started
A particle of mass 55 kg on a smooth slope of 30∘30^\circ is connected over a smooth pulley to a hanging mass MM kg. The system accelerates at 2 m s−22\ \text{m s}^{-2} with MM descending. Find MM, to 3 s.f. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
In the system of part cpi-001 (44 kg on a smooth 30∘30^\circ slope, 66 kg hanging, a=3.92 m s−2a = 3.92\ \text{m s}^{-2}), the system starts from rest. Find the speed after the masses have moved 0.50.5 m, to 3 s.f.
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A particle of mass 77 kg on a smooth slope of 25∘25^\circ is connected over a smooth pulley to a 44 kg mass hanging freely. Find the acceleration, to 3 s.f. (g=9.8g = 9.8.)
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A particle of mass 44 kg on a smooth slope of 30∘30^\circ is connected by a light string over a smooth pulley at the top to a 66 kg mass hanging freely. Find
(a) the acceleration and (b) the tension, to 3 s.f. (g=9.8g = 9.8.)mmgRT
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A particle of mass 55 kg on a rough slope of 30∘30^\circ (μ=0.2\mu = 0.2) is connected over a smooth pulley to a 44 kg mass hanging freely. Find
(a) the acceleration and (b) the tension, to 3 s.f. (g=9.8g = 9.8.)mmgRTF
●●●●●Level 34 marksStart
A particle of mass 44 kg on a rough slope of 30∘30^\circ is connected over a smooth pulley to a 55 kg hanging mass. The system accelerates at 2 m s−22\ \text{m s}^{-2}, with the 55 kg mass descending. Find the coefficient of friction, to 3 s.f. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
A particle AA of mass 66 kg lies on a rough plane inclined at 30∘30^\circ to the horizontal. The coefficient of friction between AA and the plane is 0.250.25. AA is connected by a light inextensible string, which passes over a smooth light pulley fixed at the top of the plane, to a particle BB of mass 55 kg hanging freely, as shown in the diagram. The string between AA and the pulley is parallel to a line of greatest slope of the plane. The system is released from rest with the string taut, and AA moves up the plane. (g=9.8 m s−2g = 9.8\ \text{m s}^{-2}.)AB30°6 kg5 kg(a) Show that the acceleration of the system is 0.624 m s−20.624\ \text{m s}^{-2}, correct to 3 significant figures. (5 marks)
(b) Find the tension in the string. (2 marks)
(c) When BB has descended 0.90.9 m it reaches the ground and does not rebound. Find the further distance AA moves up the plane before coming to instantaneous rest. (5 marks)
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A particle of mass 88 kg on a rough slope of 25∘25^\circ (μ=0.15\mu = 0.15) is connected over a smooth pulley to a 55 kg hanging mass. Find the acceleration, to 3 s.f. (g=9.8g = 9.8.)
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A particle of mass 33 kg on a smooth slope of 40∘40^\circ is connected over a smooth pulley to a 22 kg hanging mass. Determine whether the system moves, and if so find the acceleration, to 3 s.f. (g=9.8g = 9.8.)
●●●●●Level 32 marksStart
A particle of mass 44 kg lies on a smooth plane inclined at 30∘30^\circ to the horizontal. It is connected by a light inextensible string, over a smooth pulley at the top of the plane, to a particle of mass 33 kg hanging freely. The system is released from rest. (g=9.8 m s−2g = 9.8\ \text{m s}^{-2}.)
(a) Find the magnitude of the acceleration of the system. (4 marks)
(b) Find the tension in the string. (2 marks)
●●●●●Level 36 marksStart