SPS SPS FM Mechanics (SPS FM Mechanics) 2022 February

1. One end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N is attached to a fixed point \(O\). The other end is attached to a particle \(P\) of mass \(0.4 \mathrm {~kg} . O\) is a vertical distance of 1 m below a horizontal ceiling. \(P\) is held at a point 1.5 m vertically below \(O\) and released from rest (see diagram).
   \includegraphics[max width=\textwidth, alt={}, center]{2040be6b-3940-4ebb-83e5-8f93c2ea9688-02_480_375_370_274}

Assuming that there is no obstruction to the motion of \(P\) as it passes \(O\), find the speed of \(P\) when it first hits the ceiling.

2. A particle \(P\) of mass 2 kg is moving on a large smooth horizontal plane when it collides with a fixed smooth vertical wall. Before the collision its velocity is \(( 5 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and after the collision its velocity is \(( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. The impulse imparted on \(P\) by the wall is denoted by INs. Find the following.
   • The magnitude of \(\mathbf { I }\)
   • The angle between I and i
   • Find the loss of kinetic energy of \(P\) as a result of the collision.

1. A car has a mass of 800 kg . The engine of the car is working at a constant power of 15 kW .

In an initial model of the motion of the car it is assumed that the car is subject to a constant resistive force of magnitude \(R \mathrm {~N}\). The car is initially driven on a straight horizontal road. At the instant that its speed is \(20 \mathrm {~ms} ^ { - 1 }\) its acceleration is \(0.4 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 430\).
2. Hence find the maximum constant speed at which the car can be driven along this road, according to the initial model. In a revised model the resistance to the motion of the car at any instant is assumed to be \(60 v\) where \(v\) is the speed of the car at that instant. The car is now driven up a straight road which is inclined at an angle \(\alpha\) above the horizontal where \(\sin \alpha = 0.2\).
3. Determine the speed of the car at the instant that its acceleration is \(0.15 \mathrm {~ms} ^ { - 2 }\) up the slope, according to the revised model.

4. Fig. 5.1 shows a solid L-shaped ornament, of uniform density. The ornament is 3 cm thick. The \(x , y\) and \(z\) axes are shown, along with the dimensions of the ornament. The measurements are in centimetres. \begin{figure}[h] \end{figure}

1. Determine, with reference to the axes shown, the coordinates of the ornament's centre of mass. Fig. 5.2 shows the ornament placed so that the shaded face (indicated in Fig. 5.1) is in contact with a plane inclined at \(\theta ^ { \circ }\) to the horizontal, with the 4 cm edge parallel to a line of greatest slope. The surface of the plane is sufficiently rough so that the ornament will not slip down the plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2040be6b-3940-4ebb-83e5-8f93c2ea9688-08_597_780_1281_287} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
2. Determine the minimum and maximum possible values of \(\theta\) for which the ornament does not topple.

5. Two smooth circular discs \(A\) and \(B\) of masses \(m _ { A } \mathrm {~kg}\) and \(m _ { B } \mathrm {~kg}\) respectively are moving on a horizontal plane. At the instant before they collide the velocities of \(A\) and \(B\) are as follows, as shown in the diagram below.

• The velocity of \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha\) to the line of centres, where \(\tan \alpha = \frac { 4 } { 3 }\).
• The velocity of \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\frac { 1 } { 3 } \pi\) radians to the line of centres.
  \includegraphics[max width=\textwidth, alt={}, center]{2040be6b-3940-4ebb-83e5-8f93c2ea9688-10_442_808_443_312}

The direction of motion of \(B\) after the collision is perpendicular to the line of centres.

1. Show that \(\frac { 3 } { 2 } \leqslant \frac { m _ { B } } { m _ { A } } \leqslant 4\).
2. Given that \(m _ { A } = 2\) and \(m _ { B } = 6\), find the total loss of kinetic energy as a result of the collision.

6. A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string of length 0.8 m . The other end of the string is attached to a fixed point \(O . P\) is at rest vertically below \(O\) when it experiences a horizontal impulse of magnitude 20 Ns . In the subsequent motion the angle the string makes with the downwards vertical through \(O\) is denoted by \(\theta\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{2040be6b-3940-4ebb-83e5-8f93c2ea9688-12_373_476_365_285}

1. Find the magnitude of the acceleration of \(P\) at the first instant when \(\theta = \frac { 1 } { 3 } \pi\) radians.
2. Determine the value of \(\theta\) at which the string first becomes slack. End of Examination
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