Questions — SPS SPS FM Mechanics (26 questions)

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SPS SPS FM Mechanics 2021 January Q1
3 marks Moderate -0.5
A disc, of mass \(m\) and radius \(r\), rotates about an axis through its centre, perpendicular to the plane face of the disc. The angular speed of the disc is \(\omega\). A possible model for the kinetic energy \(E\) of the disc is $$E = km^ar^b\omega^c$$ where \(a\), \(b\) and \(c\) are constants and \(k\) is a dimensionless constant. Find the values of \(a\), \(b\) and \(c\). [3 marks]
SPS SPS FM Mechanics 2021 January Q2
11 marks Standard +0.8
The triangular region shown below is rotated through \(360°\) around the \(x\)-axis, to form a solid cone. \includegraphics{figure_1} The coordinates of the vertices of the triangle are \((0, 0)\), \((8, 0)\) and \((0, 4)\). All units are in centimetres.
  1. State an assumption that you should make about the cone in order to find the position of its centre of mass. [1 mark]
  2. Using integration, prove that the centre of mass of the cone is \(2\)cm from its plane face. [5 marks]
  3. The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding.
    1. Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree. [2 marks]
    2. Find the range of possible values for the coefficient of friction between the cone and the board. [3 marks]
SPS SPS FM Mechanics 2021 January Q3
8 marks Standard +0.8
\includegraphics{figure_2} Figure 1 represents the plan of part of a smooth horizontal floor, where \(W_1\) and \(W_2\) are two fixed parallel vertical walls. The walls are \(3\) metres apart. A particle lies at rest at a point \(O\) on the floor between the two walls, where the point \(O\) is \(d\) metres, \(0 < d \leq 3\), from \(W_1\). At time \(t = 0\), the particle is projected from \(O\) towards \(W_1\) with speed \(u\text{ms}^{-1}\) in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is \(\frac{2}{3}\). The particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.
  1. Show that \(T = \frac{45 - 5d}{4u}\). [6]
  2. The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary. Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer. [2]
SPS SPS FM Mechanics 2021 January Q4
12 marks Standard +0.3
A car of mass \(600\)kg pulls a trailer of mass \(150\)kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude \(200\)N. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car is modelled as a force of magnitude \((200 + \lambda v)\)N, where \(\lambda\) is a constant. When the engine of the car is working at a constant rate of \(15\)kW, the car is moving at a constant speed of \(25\text{ms}^{-1}\).
  1. Show that \(\lambda = 8\). [4]
  2. Later on, the car is pulling the trailer up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = \frac{1}{15}\). The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude \(200\)N at all times. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude \((200 + 8v)\)N. The engine of the car is again working at a constant rate of \(15\)kW. When \(v = 10\), the towbar breaks. The trailer comes to instantaneous rest after moving a distance \(d\) metres up the road from the point where the towbar broke. Find the acceleration of the car immediately after the towbar breaks. [4]
  3. Use the work-energy principle to find the value of \(d\). [4]
SPS SPS FM Mechanics 2021 January Q5
6 marks Standard +0.3
\includegraphics{figure_3} A hemispherical shell of radius \(a\) is fixed with its rim uppermost and horizontal. A small bead, \(B\), is moving with constant angular speed, \(\omega\), in a horizontal circle on the smooth inner surface of the shell. The centre of the path of \(B\) is at a distance \(\frac{1}{4}a\) vertically below the level of the rim of the hemisphere, as shown in Figure 1. Find the magnitude of \(\omega\), giving your answer in terms of \(a\) and \(g\). [6]
SPS SPS FM Mechanics 2021 January Q6
11 marks Challenging +1.2
Numerical (calculator) integration is not acceptable in this question. \includegraphics{figure_4} The shaded region \(OAB\) in Figure 2 is bounded by the \(x\)-axis, the line with equation \(x = 4\) and the curve with equation \(y = \frac{1}{4}(x-2)^3 + 2\). The point \(A\) has coordinates \((4, 4)\) and the point \(B\) has coordinates \((4, 0)\). A uniform lamina \(L\) has the shape of \(OAB\). The unit of length on both axes is one centimetre. The centre of mass of \(L\) is at the point with coordinates \((\bar{x}, \bar{y})\). Given that the area of \(L\) is \(8\)cm²,
  1. show that \(\bar{y} = \frac{8}{7}\). [4]
  2. The lamina is freely suspended from \(A\) and hangs in equilibrium with \(AB\) at an angle \(\theta°\) to the downward vertical. Find the value of \(\theta\). [7]
SPS SPS FM Mechanics 2021 September Q1
7 marks Moderate -0.8
A car is initially travelling with a constant velocity of \(15 \text{ m s}^{-1}\) for \(T\) s. It then decelerates at a constant rate for \(\frac{T}{2}\) s, reaching a velocity of \(10 \text{ m s}^{-1}\). It then immediately accelerates at a constant rate for \(\frac{3T}{2}\) s reaching a velocity of \(20 \text{ m s}^{-1}\).
  1. Sketch a velocity–time graph to illustrate the motion. [3]
  2. Given that the car travels a total distance of 1312.5 m over the journey described, find the value of \(T\). [4]
SPS SPS FM Mechanics 2021 September Q2
7 marks Standard +0.3
A particle \(P\) moves in a straight line. At time \(t\) s the displacement \(s\) cm from a fixed point \(O\) is given by: $$s = \frac{1}{6}\left(8t^3 - 105t^2 + 144t + 540\right).$$ Find the distance between the points at which the particle is instantaneously at rest. [7]
SPS SPS FM Mechanics 2021 September Q3
9 marks Standard +0.3
A cylindrical object with mass 8 kg rests on two cylindrical bars of equal radius. The lines connecting the centre of each of the bars to the centre of the object make an angle of \(40°\) to the vertical. \includegraphics{figure_2}
  1. Draw a diagram showing all the forces acting on the object. Describe each of the forces using words. [2]
  2. Calculate the magnitude of the force on each of the bars due to the cylindrical object. [7]
SPS SPS FM Mechanics 2021 September Q4
8 marks Standard +0.3
A box \(A\) of mass 0.8 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a smooth pulley fixed at the edge of the table. The other end of the string is attached to a sphere \(B\) of mass 1.2 kg, which hangs freely below the pulley. The magnitude of the frictional force between \(A\) and the table is \(F\) N. The system is released from rest when the string is taut. After release, \(B\) descends a distance of 0.9 m in 0.8 s. Modelling \(A\) and \(B\) as particles, calculate
  1. the acceleration of \(B\), [2]
  2. the tension in the string, [3]
  3. the value of \(F\). [3]
SPS SPS FM Mechanics 2021 September Q5
8 marks Standard +0.3
In this question use \(g = 10 \text{ m s}^{-2}\). A particle of mass 3 kg rests in limiting equilibrium on a rough plane inclined at \(30°\) to the horizontal.
  1. Find the exact value of the coefficient of friction between the particle and the plane. [2]
A horizontal force of 36 N is now applied to the particle.
  1. Find how far down the plane the particle travels after the force has been applied for 4 s. [6]
SPS SPS FM Mechanics 2022 January Q1
6 marks Standard +0.8
A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32N. The rope has natural length 4 m and modulus of elasticity of 470 N. By considering energy, determine the total distance she falls before first coming to instantaneous rest. [6]
SPS SPS FM Mechanics 2022 January Q2
5 marks Challenging +1.3
\includegraphics{figure_2} A uniform solid right circular cone has base radius \(a\) and semi-vertical angle \(\alpha\), where \(\tan \alpha = \frac{1}{3}\). The cone is freely suspended by a string attached at a point A on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of \(\theta^0\) with the upward vertical, as shown in the diagram above. Find, to one decimal place, the value of \(\theta\). [5]
SPS SPS FM Mechanics 2022 January Q3
9 marks Standard +0.3
A car of mass 800 kg is driven with its engine generating a power of 15 kW.
  1. The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds. [2]
  2. The car is next driven at constant speed up a straight road inclined at an angle \(\theta\) to the horizontal. The resistance to motion is now modelled as being constant with magnitude of 150 N. Given that \(\sin \theta = \frac{1}{20}\), find the speed of the car. [3]
  3. The car is now driven at a constant speed of 30 ms\(^{-1}\) along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming the resistance to motion of the car is three times the resistance to motion of the trailer. Find:
    1. the resistance to motion of the car,
    2. the magnitude of the tension in the towbar
    [4]
SPS SPS FM Mechanics 2022 January Q4
9 marks Challenging +1.3
\includegraphics{figure_4} Two uniform smooth spheres A and B of equal radius are moving on a horizontal surface when they collide. A has mass 0.1 kg and B has mass 0.4 kg. Immediately before the collision A is moving with speed 2.8 ms\(^{-1}\) along the line of centres, and B is moving with speed 1 ms\(^{-1}\) at an angle \(\theta\) to the line of centres, where \(\cos \theta = 0.8\) (see diagram). Immediately after the collision A is stationary. Find:
  1. the coefficient of restitution between A and B, [5]
  2. the angle turned through by the direction of motion of B as a result of the collision. [4]
SPS SPS FM Mechanics 2022 January Q5
9 marks Challenging +1.2
A right circular cone C of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of C. The other end of the string is attached to a particle P of mass 2.5 kg. P moves in a horizontal circle with constant speed and in contact with the smooth curved surface of C. The extension of the string is 1.5 m.
  1. Find the tension in the string. [2]
  2. Find the speed of P. [7]
SPS SPS FM Mechanics 2022 January Q6
8 marks Challenging +1.8
A uniform rod, PQ, of length \(2a\), rests with one end, P, on rough horizontal ground and a point T resting on a rough fixed prism of semi-circular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both P and T is \(\mu\). \includegraphics{figure_6} The rod is on the point of slipping when it is inclined at an angle of 30\(^0\) to the horizontal. Find the value of \(\mu\). [8]
SPS SPS FM Mechanics 2022 January Q7
14 marks Challenging +1.2
The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a trapezium ABCD with AB and CD perpendicular to AD. The lengths of AB and AD are each 5 cm and the length of CD is \((a + 5)\) cm. \includegraphics{figure_7}
  1. Show the distance of the centre of mass of the prism from AD is $$\frac{a^2 + 15a + 75}{3(a + 10)} \text{ cm.}$$ [5]
The prism is placed with the face containing AB in contact with a horizontal surface.
  1. Find the greatest value of \(a\) for which the prism does not topple. [3]
The prism is now placed on an inclined plane which makes an angle \(\theta^o\) with the horizontal. AB lies along a line of greatest slope with B higher than A.
  1. Using the value for \(a\) found in part (ii), and assuming the prism does not slip down the plane, find the great value of \(\theta\) for which the prism does not topple. [6]
SPS SPS FM Mechanics 2026 January Q1
8 marks Standard +0.3
A van of mass 600 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{16}\). The resistance to motion of the van from non-gravitational forces has constant magnitude \(R\) newtons. When the van is moving at a constant speed of 20 m s\(^{-1}\), the van's engine is working at a constant rate of 25 kW.
  1. Find the value of \(R\). [4]
The power developed by the van's engine is now increased to 30 kW. The resistance to motion from non-gravitational forces is unchanged. At the instant when the van is moving up the road at 20 m s\(^{-1}\), the acceleration of the van is \(a\) m s\(^{-2}\).
  1. Find the value of \(a\). [4]
SPS SPS FM Mechanics 2026 January Q2
12 marks Standard +0.3
\includegraphics{figure_2} The uniform L-shaped lamina \(OABCDE\), shown in Figure 2, is made from two identical rectangles. Each rectangle is 4 metres long and \(a\) metres wide. Giving each answer in terms of \(a\), find the distance of the centre of mass of the lamina from
  1. \(OE\). [4]
  2. \(OA\). [4]
The lamina is freely suspended from \(O\) and hangs in equilibrium with \(OE\) at an angle \(\theta\) to the downward vertical through \(O\), where \(\tan \theta = \frac{4}{3}\).
  1. Find the value of \(a\). [4]
SPS SPS FM Mechanics 2026 January Q3
8 marks Challenging +1.2
\includegraphics{figure_3} A light elastic string has natural length \(8a\) and modulus of elasticity \(5mg\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(AP = BP = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(AB\) it has speed \(\sqrt{80ag}\).
  1. Find \(L\) in terms of \(a\). [5]
  2. Find the initial acceleration of \(P\) in terms of \(g\). [3]
SPS SPS FM Mechanics 2026 January Q4
6 marks Standard +0.3
A hollow hemispherical bowl of radius \(a\) has a smooth inner surface and is fixed with its axis vertical. A particle \(P\) of mass \(m\) moves in horizontal circles on the inner surface of the bowl, at a height \(x\) above the lowest point of the bowl. The speed of \(P\) is \(\sqrt{\frac{g}{2}a}\). Find \(x\) in terms of \(a\). [6]
SPS SPS FM Mechanics 2026 January Q5
8 marks Challenging +1.2
\includegraphics{figure_5} \(AB\) and \(BC\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(ABC = 60°\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(AB\) at an angle \(\theta\) with \(AB\). It then strikes \(BC\) and rebounds at an angle \(\beta\) with \(BC\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is 40% of its kinetic energy before the first collision.
  1. Find the value of \(e\). [4]
  2. Find the size of angle \(\beta\). [4]
SPS SPS FM Mechanics 2026 January Q6
8 marks Challenging +1.2
\includegraphics{figure_6} A particle \(P\) of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m. The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass 0.04 kg is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius 0.8 m with angular speed \(\omega\) rad s\(^{-1}\). The particle \(Q\) moves in a horizontal circle of radius 1.4 m also with angular speed \(\omega\) rad s\(^{-1}\). The centres of the circles are vertically below \(O\), and \(O\), \(P\) and \(Q\) are always in the same vertical plane. The strings \(OP\) and \(PQ\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).
  1. Find the tension in the string \(OP\). [3]
  2. Find the value of \(\omega\). [3]
  3. Find the value of \(\beta\). [2]
SPS SPS FM Mechanics 2026 January Q7
9 marks Challenging +1.2
\includegraphics{figure_7} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac{1}{2}m\) respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere \(A\) is travelling with speed \(u\) and its direction of motion makes an angle \(\alpha\) with the line of centres. Sphere \(B\) is travelling with speed \(2u\) and its direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{2}{3}\) and \(\alpha + \beta = 90°\).
  1. Find the component of the velocity of \(B\) parallel to the line of centres after the collision, giving your answer in terms of \(u\) and \(\alpha\). [4]
The direction of motion of \(B\) after the collision is parallel to the direction of motion of \(A\) before the collision.
  1. Find the value of \(\tan \alpha\). [5]