OCR Further Mechanics AS (Further Mechanics AS) 2024 June

1 A particle \(P\) of mass 2.5 kg is moving with a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal plane when it collides directly with a fixed vertical wall. After the collision \(P\) moves away from the wall with a speed of \(2.8 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the coefficient of restitution between \(P\) and the wall.
2. Find the magnitude and state the direction of the impulse exerted on \(P\) by the wall.
3. State the magnitude and direction of the impulse exerted on the wall by \(P\).

2 A particle \(P\) of mass 0.4 kg is attached to one end of a light inextensible string of length 1.8 m . The other end of the string is attached to a fixed point, \(O\), on a smooth horizontal plane. Initially, \(P\) is moving with a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) in a horizontal circle with \(O\) as its centre.

1. 1. Find the magnitude of the acceleration of \(P\).
   2. State the direction of the acceleration of \(P\). A force is now applied to \(P\) in such a way that its angular velocity increases. At the instant that the angular velocity reaches \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\), the string breaks.
2. 1. Find the speed with which \(P\) is moving at the instant that the string breaks.
   2. Find the tension in the string at the instant that the string breaks. After the string has broken \(P\) starts to move directly up a smooth slope which is fixed to the plane and inclined at an angle \(\theta ^ { \circ }\) above the horizontal. Particle \(P\) moves a distance of 20 m up the slope before coming to instantaneous rest.
3. Use an energy method to determine the value of \(\theta\).

3 A small object \(P\) of mass \(m\) is suspended from a fixed point by a light inextensible string of length l. When \(P\) is displaced and released in a certain way, it oscillates in a vertical plane. The time taken for one complete oscillation is called the period and is denoted by \(\tau\). A student is carrying out experiments with \(P\) and suggests the following formula to model the value of \(\tau\).
\(\tau = \mathrm { cg } \mathrm { a } ^ { \mathrm { a } } \mathrm { l } _ { \mathrm { m } } { } ^ { \gamma }\)
in which

• \(g\) is the acceleration due to gravity,
• \(C\) is a dimensionless constant.
  1. Use dimensional analysis to determine the values of the constants \(\alpha , \beta\) and \(\gamma\).
  2. 1. Determine the effect on the period, according to the model, if the length of the string is then multiplied by 4, all other conditions being unchanged.
     2. Determine the effect on the period, according to the model, if instead the mass of the object is multiplied by 4, all other conditions being unchanged.

4 A particle \(B\) of mass 5 kg is at rest at the bottom of a slope which is angled at \(\sin ^ { - 1 } 0.2\) above the horizontal. A constant force \(D\) initially acts directly up the slope on \(B\). The total resistance to the motion of \(B\) is modelled as being a constant 12 N .
At the instant that \(D\) stops acting, the speed of \(B\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has moved 90 m up the slope.
Determine the average power of \(D\) over the time that \(D\) has been acting on \(B\).

5 Two particles, \(A\) of mass \(m _ { A } \mathrm {~kg}\) and \(B\) of mass 5 kg , are moving directly towards each other on a smooth horizontal floor. Before they collide they have speeds \(\mathrm { u } _ { \mathrm { A } } \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Immediately after they collide the direction of motion of each particle has been reversed and \(A\) and \(B\) have speeds \(3.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). The coefficient of restitution between \(A\) and \(B\) is 0.75 . Before:
\includegraphics[max width=\textwidth, alt={}, center]{d2156252-71f2-4084-89a2-4d246583eb65-4_218_711_552_283} After:
\includegraphics[max width=\textwidth, alt={}, center]{d2156252-71f2-4084-89a2-4d246583eb65-4_218_707_552_1078}

1. Determine the value of \(m _ { A }\) and the value of \(u _ { A }\).
   [0pt] [5]
2. Show that approximately \(41 \%\) of the kinetic energy of the system is lost in this collision. After the collision between \(A\) and \(B\), \(B\) goes on to collide directly with a third particle \(C\) of mass 3 kg which is travelling towards \(B\) with a speed of \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between \(B\) and \(C\) is denoted by \(e\).
3. Given that, after \(B\) and \(C\) collide, there are no further collisions between \(A , B\) and \(C\) determine the range of possible values of \(e\).

6 A motorbike and its rider, together denoted by \(M\), have a combined mass of 360 kg . The resistive force experienced by \(M\) when it is in motion is modelled as being proportional to the speed it is moving at. All motion of \(M\) is on a straight horizontal road. It is found that with the engine of the motorbike working at a rate of 12 kW , the maximum constant speed that \(M\) can move at is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Determine the speed of \(M\) such that with the engine working at a rate of 12 kW the acceleration of \(M\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

7 A particle \(P\) of mass 3.5 kg is attached to one end of a rod of length 5.4 m . The other end of the rod is hinged at a fixed point \(O\) and \(P\) hangs in equilibrium directly below \(O\). A horizontal impulse of magnitude 44.1 Ns is applied to \(P\).
In an initial model of the subsequent motion of \(P\) the rod is modelled as being light and inextensible and all resistance to the motion of \(P\) is ignored. You are given that \(P\) moves in a circular path in a vertical plane containing \(O\). The angle that the rod makes with the downward vertical through \(O\) is \(\theta\) radians.

1. Determine the largest value of \(\theta\) in the subsequent motion of \(P\). In a revised model the rod is still modelled as being light and inextensible but the resistance to the motion of \(P\) is not ignored. Instead, it is modelled as causing a loss of energy of 20 J for every metre that \(P\) travels.
2. Show that according to the revised model, the maximum value of \(\theta\) in the subsequent motion of \(P\) satisfies the following equation. $$343 ( 1 + 2 \cos \theta ) = 400 \theta$$ You are given that \(\theta = 1.306\) is the solution to the above equation, correct to \(\mathbf { 4 }\) significant figures.
3. Determine the difference in the predicted maximum vertical heights attained by \(P\) using the two models. Give your answer correct to \(\mathbf { 3 }\) significant figures.
4. Suggest one further improvement that could be made to the model of the motion of \(P\).