Edexcel M1 (Mechanics 1) 2021 October

1. \begin{figure}[h] \end{figure} A non-uniform rod \(A B\) has length 9 m and mass \(M \mathrm {~kg}\).
The rod rests in equilibrium in a horizontal position on two supports, one at \(C\) where \(A C = 2.5 \mathrm {~m}\) and the other at \(D\) where \(D B = 2 \mathrm {~m}\), as shown in Figure 1 . The magnitude of the force acting on the rod at \(D\) is twice the magnitude of the force acting on the \(\operatorname { rod }\) at \(C\). The centre of mass of the rod is \(d\) metres from \(A\).
Find the value of \(d\).

2. A particle \(P\) of mass \(2 m\) is moving on a rough horizontal plane when it collides directly with a particle \(Q\) of mass \(4 m\) which is at rest on the plane. The speed of \(P\) immediately before the collision is \(3 u\). The speed of \(Q\) immediately after the collision is \(2 u\).

1. Find, in terms of \(u\), the speed of \(P\) immediately after the collision.
2. State clearly the direction of motion of \(P\) immediately after the collision. Following the collision, \(Q\) comes to rest after travelling a distance \(\frac { 6 u ^ { 2 } } { g }\) along the plane. The coefficient of friction between \(Q\) and the plane is \(\mu\).
3. Find the value of \(\mu\).

3. A car is moving at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. The car is modelled as a particle.
At time \(t = 0\), the car is at the point \(A\) and the driver sees a road sign 48 m ahead.
Let \(t\) seconds be the time that elapses after the car passes \(A\).
In a first model, the car is assumed to decelerate uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) from \(A\) until the car reaches the road sign.

1. Use this first model to find the speed of the car as it reaches the sign. The road sign indicates that the speed limit immediately after the sign is \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   In a second model, the car is assumed to decelerate uniformly at \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from \(A\) until it reaches a speed of \(13 \mathrm {~ms} ^ { - 1 }\). The car then maintains this speed until it reaches the road sign.
2. Use this second model to find the value of \(t\) at which the car reaches the sign. In a third model, the car is assumed to move with constant speed \(25 \mathrm {~ms} ^ { - 1 }\) from \(A\) until time \(t = 0.2\), the car then decelerates uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) until it reaches a speed of \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then maintains this speed until it reaches the road sign.
3. Use this third model to find the value of \(t\) at which the car reaches the sign.

1. The position vector, \(\mathbf { r }\) metres, of a particle \(P\) at time \(t\) seconds, relative to a fixed origin \(O\), is given by

$$\mathbf { r } = ( t - 3 ) \mathbf { i } + ( 1 - 2 t ) \mathbf { j }$$

1. Find, to the nearest degree, the size of the angle between \(\mathbf { r }\) and the vector \(\mathbf { j }\), when \(t = 2\)
2. Find the values of \(t\) for which the distance of \(P\) from \(O\) is 2.5 m .

5. \begin{figure}[h] \end{figure} A small bead of mass 0.2 kg is attached to the end \(P\) of a light rod \(P Q\). The bead is threaded onto a fixed vertical rough wire. The bead is held in equilibrium with the \(\operatorname { rod } P Q\) inclined to the wire at an angle \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 2. The thrust in the rod is \(T\) newtons.
The bead is modelled as a particle.

1. Find the magnitude and direction of the friction force acting on the bead when \(T = 2.5\) The coefficient of friction between the bead and the wire is \(\mu\).
   Given that the greatest possible value of \(T\) is 6.125
2. find the value of \(\mu\).

6. \begin{figure}[h] \end{figure} A small ball is thrown vertically upwards at time \(t = 0\) from a point \(A\) which is above horizontal ground. The ball hits the ground 7 s later. The ball is modelled as a particle moving freely under gravity.
The velocity-time graph shown in Figure 3 represents the motion of the ball for \(0 \leqslant t \leqslant 7\)

1. Find the speed with which the ball is thrown.
2. Find the height of \(A\) above the ground.

7. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a particle \(A\) of mass \(2 m\). The other end of the string is attached to a particle \(B\) of mass \(3 m\). The string passes over a small, smooth, light pulley \(P\) which is fixed at the top of a rough inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) Particle \(A\) is held at rest on the plane with the string taut and \(B\) hanging freely below \(P\), as shown in Figure 4. The section of the string \(A P\) is parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 2 }\)
Particle \(A\) is released and begins to move up the plane.
For the motion before \(A\) reaches the pulley,

1. 1. write down an equation of motion for \(A\),
   2. write down an equation of motion for \(B\),
2. find, in terms of \(g\), the acceleration of \(A\),
3. find the magnitude of the force exerted on the pulley by the string.
4. State how you have used the information that \(P\) is a smooth pulley.

8. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin.] At 7 am a ship leaves a port and moves with constant velocity. The position vector of the port is \(( - 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { km }\). At 7.36 am the ship is at the point with position vector \(( 4 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km }\).

1. Show that the velocity of the ship is \(( 10 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)
2. Find the position vector of the ship \(t\) hours after leaving port. At 8.48 am a passenger on the ship notices that a lighthouse is due east of the ship. At 9 am the same passenger notices that the lighthouse is now north east of the ship.
3. Find the position vector of the lighthouse.
4. Find the position vector of the ship when it is due south of the lighthouse.
   
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