SPS SPS SM Mechanics (SPS SM Mechanics) 2026 January

1. A particle is thrown vertically upwards and returns to its point of projection after 6 seconds. Air resistance is negligible.

Calculate the speed of projection of the particle and also the maximum height it reaches. \includegraphics[max width=\textwidth, alt={}, center]{fb36606d-0ee3-4050-af31-1642e5f67a03-05_2688_1886_118_118}

2. The resultant of the force \(\binom { - 4 } { 8 } \mathrm {~N}\) and the force \(\mathbf { F }\) gives an object of mass 6 kg an acceleration of \(\binom { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Calculate \(\mathbf { F }\).
2. Calculate the angle between \(\mathbf { F }\) and the vector \(\binom { 0 } { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{fb36606d-0ee3-4050-af31-1642e5f67a03-07_2688_1886_118_118}

3. A man of mass 75 kg is standing in a lift. He is holding a parcel of mass 5 kg by means of a light inextensible string, as shown in Fig. 5. The tension in the string is 55 N . \begin{figure}[h] \end{figure}

1. Find the upward acceleration.
2. Find the reaction on the man of the lift floor. \includegraphics[max width=\textwidth, alt={}, center]{fb36606d-0ee3-4050-af31-1642e5f67a03-09_2688_1886_118_118}

4. A toy car is travelling in a straight horizontal line. One model of the motion for \(0 \leqslant t \leqslant 8\), where \(t\) is the time in seconds, is shown in the velocity-time graph Fig. 6. \begin{figure}[h] \end{figure}

1. Calculate the distance travelled by the car from \(t = 0\) to \(t = 8\).
2. How much less time would the car have taken to travel this distance if it had maintained its initial speed throughout?
3. What is the acceleration of the car when \(t = 1\) ? From \(t = 8\) to \(t = 14\), the car travels 58.5 m with a new constant acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
4. Find \(a\). A second model for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the toy car is $$v = 12 - 10 t + \frac { 9 } { 4 } t ^ { 2 } - \frac { 1 } { 8 } t ^ { 3 } , \text { for } 0 \leqslant t \leqslant 8$$ This model agrees with the values for \(v\) given in Fig. 6 for \(t = 0,2,4\) and 6. [Note that you are not required to verify this.] Use this second model to answer the following questions.
5. Calculate the acceleration of the car when \(t = 1\).
6. Initially the car is at A. Find an expression in terms of \(t\) for the displacement of the car from A after the first \(t\) seconds of its motion. Hence find the displacement of the car from A when \(t = 8\). \includegraphics[max width=\textwidth, alt={}, center]{fb36606d-0ee3-4050-af31-1642e5f67a03-13_2688_1886_118_118}

5. A toy sledge of mass 4 kg is being pulled in a straight line by a light string. The resistance to its motion is 6 N . \begin{figure}[h] \end{figure} At one time, the string is horizontal and the sledge is on horizontal ground, as shown in Fig. 6.1. The acceleration of the sledge is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) forwards.

1. Calculate the tension in the string. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fb36606d-0ee3-4050-af31-1642e5f67a03-14_190_718_813_733} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure} At another time, the sledge is again on horizontal ground but the string is now at \(40 ^ { \circ }\) to the horizontal, as shown in Fig. 6.2. The tension in the string is 25 N .
2. Calculate the acceleration of the sledge. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fb36606d-0ee3-4050-af31-1642e5f67a03-16_364_465_283_479} \captionsetup{labelformat=empty} \caption{Fig. 6.3}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fb36606d-0ee3-4050-af31-1642e5f67a03-16_414_463_233_1226} \captionsetup{labelformat=empty} \caption{Fig. 6.4}
   \end{figure} In another situation the sledge is on a slope inclined at \(35 ^ { \circ }\) to the horizontal, as shown in Fig. 6.3. It is held in equilibrium by the light string parallel to the slope. The resistance to motion of 6 N acts up the slope.
3. Calculate the tension in the string. The sledge is now held in equilibrium with the light string inclined at \(\theta ^ { \circ }\) to the slope, as shown in Fig. 6.4. The tension in the string is 25 N and the resistance to motion remains 6 N acting up the slope.
4. (A) Show all the forces acting on the sledge.
   (B) Calculate the angle \(\theta\).
   (C) Calculate the normal reaction of the slope on the sledge. \includegraphics[max width=\textwidth, alt={}, center]{fb36606d-0ee3-4050-af31-1642e5f67a03-17_2688_1886_118_118}

6. A box of weight 147 N is held by light strings AB and BC . As shown in Fig. 7.1, AB is inclined at \(\alpha\) to the horizontal and is fixed at \(\mathrm { A } ; \mathrm { BC }\) is held at C . The box is in equilibrium with BC horizontal and \(\alpha\) such that \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). \begin{figure}[h] \end{figure}

1. Calculate the tension in string AB .
2. Show that the tension in string BC is 196 N . As shown in Fig. 7.2, a box of weight 90 N is now attached at C and another light string CD is held at D so that the system is in equilibrium with BC still horizontal. CD is inclined at \(\beta\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fb36606d-0ee3-4050-af31-1642e5f67a03-20_394_714_340_671} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure}
3. Explain why the tension in the string BC is still 196 N .
4. Draw a diagram showing the forces acting on the box at C . Find the angle \(\beta\) and show that the tension in CD is 216 N , correct to three significant figures. The string section CD is now taken over a smooth pulley and attached to a block of mass \(M \mathrm {~kg}\) on a rough slope inclined at \(40 ^ { \circ }\) to the horizontal. As shown in Fig. 7.3, the part of the string attached to the box is still at \(\beta\) to the horizontal and the part attached to the block is parallel to the slope. The system is in equilibrium with a frictional force of 20 N acting on the block up the slope. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fb36606d-0ee3-4050-af31-1642e5f67a03-22_451_1070_422_495} \captionsetup{labelformat=empty} \caption{Fig. 7.3}
   \end{figure}
5. Calculate the value of \(M\). \section*{End of Examination} [BLANK PAGE]
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