3.01b Derived quantities and units

6 \includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-3_504_878_1557_632} One end of a light inextensible string of length 0.7 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass 0.25 kg . The particle \(P\) moves in a circle on a smooth horizontal table with constant speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The string is taut and makes an angle of \(40 ^ { \circ }\) with the vertical (see diagram). Find

1. the tension in the string,
2. the force exerted on \(P\) by the table. \(P\) now moves in the same horizontal circle with constant angular speed \(\omega \operatorname { rad~s } ^ { - 1 }\).
3. Find the maximum value of \(\omega\) for which \(P\) remains on the table.

1 A particle moves under the action of a force, \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), of the particle is given by $$\mathbf { v } = \left( t ^ { 3 } - 15 t - 5 \right) \mathbf { i } + \left( 6 t - t ^ { 2 } \right) \mathbf { j }$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The mass of the particle is 4 kg .
   1. Show that, at time \(t\), $$\mathbf { F } = \left( 12 t ^ { 2 } - 60 \right) \mathbf { i } + ( 24 - 8 t ) \mathbf { j }$$
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 2\).

A single force of magnitude 4 newtons acts on a particle of mass 50 grams. Find the magnitude of the acceleration of the particle. Circle your answer. [1 mark] \(12.5 \text{ m s}^{-2}\) \(\quad\) \(0.08 \text{ m s}^{-2}\) \(\quad\) \(0.0125 \text{ m s}^{-2}\) \(\quad\) \(80 \text{ m s}^{-2}\)

The graph below models the velocity of a small train as it moves on a straight track for 20 seconds. The front of the train is at the point \(A\) when \(t = 0\) The mass of the train is 800kg. \includegraphics{figure_14}

1. Find the total distance travelled in the 20 seconds. [3 marks]
2. Find the distance of the front of the train from the point \(A\) at the end of the 20 seconds. [1 mark]
3. Find the maximum magnitude of the resultant force acting on the train. [2 marks]
4. Explain why, in reality, the graph may not be an accurate model of the motion of the train. [1 mark]