Questions M1 (1912 questions)

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Edexcel M1 2022 January Q4
8 marks Moderate -0.3
4. At time \(t = 0\), a small ball is projected vertically upwards from a point \(A\) which is 24.5 m above the ground. The ball first comes to instantaneous rest at the point \(B\), where \(A B = 19.6 \mathrm {~m}\) and first hits the ground at time \(t = T\) seconds. The ball is modelled as a particle moving freely under gravity.
  1. Find the value of \(T\).
  2. Sketch a speed-time graph for the motion of the ball from \(t = 0\) to \(t = T\) seconds.
    (No further calculations are needed in order to draw this sketch.)
Edexcel M1 2022 January Q5
8 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-14_209_511_246_721} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle of mass \(m\) rests in equilibrium on a fixed rough plane under the action of a force of magnitude \(X\). The force acts up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\)
The coefficient of friction between the particle and the plane is \(\mu\).
  • When \(X = 2 P\), the particle is on the point of sliding up the plane.
  • When \(X = P\), the particle is on the point of sliding down the plane.
Find the value of \(\mu\).
Edexcel M1 2022 January Q6
12 marks Moderate -0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.]
A particle \(P\) of mass 2 kg moves under the action of two forces, \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\) and \(( 2 q \mathbf { i } + p \mathbf { j } ) \mathrm { N }\), where \(p\) and \(q\) are constants. Given that the acceleration of \(P\) is \(( \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 2 }\)
  1. find the value of \(p\) and the value of \(q\).
  2. Find the size of the angle between the direction of the acceleration and the vector \(\mathbf { j }\). At time \(t = 0\), the velocity of \(P\) is \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
    At \(t = T\) seconds, \(P\) is moving in the direction of the vector \(( 11 \mathbf { i } - 13 \mathbf { j } )\).
  3. Find the value of \(T\).
Edexcel M1 2022 January Q7
13 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-22_342_1203_246_374} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle \(P\) of mass \(4 m\) lies on the surface of a fixed rough inclined plane.
The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\)
The particle \(P\) is attached to one end of a light inextensible string.
The string passes over a small smooth pulley that is fixed at the top of the plane. The other end of the string is attached to a particle \(Q\) of mass \(m\) which lies on a smooth horizontal plane. The string lies along the horizontal plane and in the vertical plane that contains the pulley and a line of greatest slope of the inclined plane. The system is released from rest with the string taut, as shown in Figure 4, and \(P\) moves down the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\)
For the motion before \(Q\) reaches the pulley
  1. write down an equation of motion for \(Q\),
  2. find, in terms of \(m\) and \(g\), the tension in the string,
  3. find the magnitude of the force exerted on the pulley by the string.
  4. State where in your working you have used the information that the string is light.
Edexcel M1 2022 January Q8
14 marks Standard +0.3
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.] A ship \(A\) moves with constant velocity \(( 3 \mathbf { i } - 10 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\)
At time \(t\) hours, the position vector of \(A\) is \(\mathbf { r } \mathrm { km }\).
At time \(t = 0 , A\) is at the point with position vector \(( 13 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\).
  1. Find \(\mathbf { r }\) in terms of \(t\). Another ship \(B\) moves with constant velocity \(( 15 \mathbf { i } + 14 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)
    At time \(t = 0 , B\) is at the point with position vector \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km }\).
  2. Show that, at time \(t\) hours, $$\overrightarrow { A B } = [ ( 12 t - 10 ) \mathbf { i } + ( 24 t - 10 ) \mathbf { j } ] \mathrm { km }$$ Given that the two ships do not change course,
  3. find the shortest distance between the two ships,
  4. find the bearing of ship \(B\) from ship \(A\) when the ships are closest.
    \includegraphics[max width=\textwidth, alt={}]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-28_2820_1967_102_100}
Edexcel M1 2023 January Q1
10 marks Moderate -0.3
  1. A train travels along a straight horizontal track between two stations \(A\) and \(B\).
The train starts from rest at station \(A\) and accelerates uniformly for \(T\) seconds until it reaches a speed of \(20 \mathrm {~ms} ^ { - 1 }\) The train then travels at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 3 minutes before decelerating uniformly until it comes to rest at station \(B\). The magnitude of the acceleration of the train is twice the magnitude of the deceleration.
  1. On the axes below, sketch a speed-time graph to illustrate the motion of the train as it moves from station \(A\) to station \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{84c0eead-0a87-4d87-b33d-794a94bb466c-02_670_1422_813_312} If you need to redraw your graph, use the axes on page 3 Stations \(A\) and \(B\) are 4.8 km apart.
  2. Find the value of \(T\)
  3. Find the acceleration of the train during the first \(T\) seconds of its motion. Only use these axes if you need to redraw your graph. \({ } _ { O } ^ { \substack { \text { speed } \\ \left( \mathrm { ms } ^ { - 1 } \right) } }\)
Edexcel M1 2023 January Q2
8 marks Standard +0.3
2. Two particles, \(A\) and \(B\), are moving in a straight line in opposite directions towards each other on a smooth horizontal surface when they collide directly. Particle \(A\) has mass \(3 m \mathrm {~kg}\) and particle \(B\) has mass \(m \mathrm {~kg}\).
Immediately before the collision, both particles have a speed of \(1.5 \mathrm {~ms} ^ { - 1 }\)
Immediately after the collision, the direction of motion of \(A\) is unchanged and the difference between the speed of \(A\) and speed of \(B\) is \(1 \mathrm {~ms} ^ { - 1 }\)
  1. Find (i) the speed of \(A\) immediately after the collision,
    (ii) the speed of \(B\) immediately after the collision.
  2. Find, in terms of \(m\), the magnitude of the impulse exerted on \(B\) in the collision.
Edexcel M1 2023 January Q3
10 marks Moderate -0.8
  1. A particle \(P\) is moving with constant acceleration ( \(- 4 \mathbf { i } + \mathbf { j }\) ) \(\mathrm { ms } ^ { - 2 }\)
At time \(t = 0 , P\) has velocity \(( 14 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
  1. Find the speed of \(P\) at time \(t = 2\) seconds.
  2. Find the size of the angle between the direction of \(\mathbf { i }\) and the direction of motion of \(P\) at time \(t = 2\) seconds. At time \(t = T\) seconds, \(P\) is moving in the direction of vector ( \(2 \mathbf { i } - 3 \mathbf { j }\) )
  3. Find the value of \(T\)
Edexcel M1 2023 January Q4
8 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{84c0eead-0a87-4d87-b33d-794a94bb466c-10_419_1445_283_312} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A branch \(A B\), of length 1.5 m , rests horizontally in equilibrium on two supports.
The two supports are at the points \(C\) and \(D\), where \(A C = 0.24 \mathrm {~m}\) and \(D B = 0.36 \mathrm {~m}\), as shown in Figure 1. When a force of 150 N is applied vertically upwards at \(B\), the branch is on the point of tilting about \(C\). When a force of 225 N is applied vertically downwards at \(B\), the branch is on the point of tilting about \(D\). The branch is modelled as a non-uniform rod \(A B\) of weight \(W\) newtons.
The distance from the point \(C\) to the centre of mass of the rod is \(x\) metres.
Use the model to find
  1. the value of \(W\)
  2. the value of \(x\)
Edexcel M1 2023 January Q5
9 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{84c0eead-0a87-4d87-b33d-794a94bb466c-14_117_1393_328_337} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Three points \(P , Q\) and \(R\) are on a horizontal road where \(P Q R\) is a straight line.
The point \(Q\) is between \(P\) and \(R\), with \(P Q = 6 x\) metres and \(Q R = 5 x\) metres, as shown in Figure 2. A vehicle moves along the road from \(P\) to \(Q\) with constant acceleration.
The vehicle is modelled as a particle.
At time \(t = 0\), the vehicle passes \(P\) with speed \(u \mathrm {~ms} ^ { - 1 }\)
At time \(t = 12 \mathrm {~s}\), the vehicle passes \(Q\) with speed \(2 u \mathrm {~ms} ^ { - 1 }\)
Using the model,
  1. show that \(x = 3 u\) As the vehicle passes \(Q\), the acceleration of the vehicle changes instantaneously to \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The vehicle continues to move with a constant acceleration of \(1.5 \mathrm {~ms} ^ { - 2 }\) and passes \(R\) with speed \(3 u \mathrm {~ms} ^ { - 1 }\) Using the model,
  2. find the value of \(u\),
  3. find the distance travelled by the vehicle during the first 14 seconds after passing \(P\)
Edexcel M1 2023 January Q6
8 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{84c0eead-0a87-4d87-b33d-794a94bb466c-18_502_1429_280_319} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A boat is pulled along a river at a constant speed by two ropes.
The banks of the river are parallel and the boat travels horizontally in a straight line, parallel to the riverbanks.
  • The tension in the first rope is 500 N acting at an angle of \(40 ^ { \circ }\) to the direction of motion, as shown in Figure 3.
  • The tension in the second rope is \(P\) newtons, acting at an angle of \(\alpha ^ { \circ }\) to the direction of motion, also shown in Figure 3.
  • The resistance to motion of the boat as it moves through the water is a constant force of magnitude 900 N
The boat is modelled as a particle. The ropes are modelled as being light and lying in a horizontal plane. Use the model to find
  1. the value of \(\alpha\)
  2. the value of \(P\)
Edexcel M1 2023 January Q7
7 marks Moderate -0.5
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{84c0eead-0a87-4d87-b33d-794a94bb466c-22_341_316_283_877} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A simple lift operates by means of a vertical cable which is attached to the top of the lift. The lift has mass \(m\)
A box \(Q\) is placed on the floor of the lift.
A box \(P\) is placed directly on top of box \(Q\), as shown in Figure 4.
The cable is modelled as being light and inextensible and air resistance is modelled as being negligible.
The tension in the cable is \(\frac { 42 m g } { 5 }\)
The lift and its contents move vertically upwards with acceleration \(\frac { 2 g } { 5 }\)
Using the model,
  1. find, in terms of \(m\), the combined mass of boxes \(P\) and \(Q\) During the motion of the lift, the force exerted on box \(P\) by box \(Q\) is \(\frac { 14 m g } { 5 }\) Using the model,
  2. find, in terms of \(m\), the mass of box \(P\)
Edexcel M1 2023 January Q8
15 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{84c0eead-0a87-4d87-b33d-794a94bb466c-24_545_764_285_651} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A parcel of mass 2 kg is pulled up a rough inclined plane by the action of a constant force. The force has magnitude 18 N and acts at an angle of \(40 ^ { \circ }\) to the plane.
The line of action of the force lies in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 5.
The coefficient of friction between the plane and the parcel is 0.3
The parcel is modelled as a particle \(P\)
  1. Find the acceleration of \(P\) The points \(A\) and \(B\) lie on a line of greatest slope of the plane, where \(A B = 5 \mathrm {~m}\) and \(B\) is above \(A\). Particle \(P\) passes through \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(A B\).
  2. Find the speed of \(P\) as it passes through \(B\). The force of 18 N is removed at the instant \(P\) passes through \(B\). As a result, \(P\) comes to rest at the point \(C\).
  3. Determine whether \(P\) will remain at rest at \(C\). You must show all stages of your working clearly.
Edexcel M1 2024 January Q1
6 marks Moderate -0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e59a66b8-c2ad-41fd-9959-9d21e9455c37-02_438_1374_246_347} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a small smooth ring threaded onto a light inextensible string.
One end of the string is attached to a fixed point \(A\) on a horizontal ceiling and the other end of the string is attached to a fixed point \(B\) on the ceiling. A horizontal force of magnitude 2 N acts on the ring so that the ring rests in equilibrium at a point \(C\), vertically below \(B\), with the string taut. The line of action of the horizontal force and the string both lie in the same vertical plane. The angle that the string makes with the ceiling at \(A\) is \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\)
The tension in the string is \(T\) newtons. The mass of the ring is \(M \mathrm {~kg}\).
  1. Find the value of \(T\)
  2. Find the value of \(M\)
Edexcel M1 2024 January Q2
6 marks Moderate -0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e59a66b8-c2ad-41fd-9959-9d21e9455c37-04_204_947_242_559} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows two particles, \(A\) and \(B\), moving in opposite directions on a smooth horizontal surface. Particle \(A\) has mass 5 kg and particle \(B\) has mass \(x \mathrm {~kg}\). The particles collide directly.
Immediately before the collision, the speed of \(A\) is \(3 \mathrm {~ms} ^ { - 1 }\) and the speed of \(B\) is \(x \mathrm {~ms} ^ { - 1 }\)
Immediately after the collision, the speed of \(A\) is \(1 \mathrm {~ms} ^ { - 1 }\) and its direction of motion is unchanged. Immediately after the collision, the speed of \(B\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  1. Find the value of \(x\).
  2. Find the magnitude of the impulse exerted on \(A\) by \(B\) in the collision.
Edexcel M1 2024 January Q3
10 marks Moderate -0.3
  1. A van travels with constant acceleration along a straight horizontal road.
The van passes a point \(A\) with speed \(u \mathrm {~ms} ^ { - 1 }\) and 20 seconds later passes a point \(B\) with speed \(28 \mathrm {~ms} ^ { - 1 }\) The distance \(A B\) is 400 m .
  1. Show that \(u = 12\)
  2. Find the time taken for the van to travel from \(A\) to the midpoint of \(A B\). The van has mass 1200 kg .
    During its motion the van experiences a constant resistive force of magnitude 260 N
  3. Find the magnitude of the driving force exerted by the engine of the van as it travels from \(A\) to \(B\).
Edexcel M1 2024 January Q4
8 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e59a66b8-c2ad-41fd-9959-9d21e9455c37-08_399_889_246_587} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) acting on a particle.
The angle between the direction of \(\mathbf { P }\) and the direction of \(\mathbf { Q }\) is \(150 ^ { \circ }\)
Force \(\mathbf { P }\) has magnitude \(X\) newtons.
Force \(\mathbf { Q }\) has magnitude \(5 \sqrt { 3 } \mathrm {~N}\).
The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) has magnitude \(\sqrt { 129 } \mathrm {~N}\).
Find
  1. the value of \(X\).
  2. the angle between \(\mathbf { Q }\) and the resultant, giving your answer to the nearest degree.
Edexcel M1 2024 January Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e59a66b8-c2ad-41fd-9959-9d21e9455c37-12_412_1529_242_267} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A beam \(A B\) has mass 30 kg and length 3 m .
The beam rests on supports at \(C\) and \(D\) where \(A C = 0.4 \mathrm {~m}\) and \(D B = 0.4 \mathrm {~m}\), as shown in Figure 4. A person of mass 55 kg stands on the beam between \(C\) and \(D\).
The person is modelled as a particle at the point \(P\), where \(C P = x\) metres and \(0 < x < 2.2\) The beam is modelled as a uniform rod resting in equilibrium in a horizontal position.
Using the model,
  1. show that the magnitude of the reaction at \(C\) is \(( 686 - 245 x ) \mathrm { N }\). The magnitude of the reaction at \(C\) is four times the magnitude of the reaction at \(D\).
    Using the model,
  2. find the value of \(x\) The person steps off the beam and places a package of mass \(M \mathrm {~kg}\) at \(A\).
    The package is modelled as a particle at the point \(A\).
    The beam is now on the point of tilting about \(C\).
    Using the model,
  3. find the value of \(M\)
Edexcel M1 2024 January Q6
12 marks Moderate -0.8
  1. A particle is projected vertically upwards from a point \(A\) with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The point \(A\) is 2.5 m vertically above the point \(B\).
Point \(B\) lies on horizontal ground.
The particle moves freely under gravity until it hits the ground at \(B\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) After hitting the ground the particle does not rebound.
  1. Find the value of \(V\).
  2. Find the time taken for the particle to reach \(B\). The point \(C\) is 10 m vertically above \(A\).
  3. Find the length of time for which the particle is above \(C\).
  4. Sketch a speed-time graph for the motion of the particle from projection to the instant that it reaches \(B\). (No further calculations are required.)
Edexcel M1 2024 January Q7
11 marks Standard +0.3
  1. \hspace{0pt} [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 \(O\).]
At midnight, a ship \(S\) is at the point with position vector ( \(19 \mathbf { i } + 22 \mathbf { j }\) )km
The ship travels with constant velocity \(( 12 \mathbf { i } - 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)
  1. Find the speed of \(S\). At time \(t\) hours after midnight, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\).
  2. Find an expression for \(\mathbf { s }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(t\). A lighthouse stands on a small rocky island. The lighthouse is modelled as being at the point with position vector \(( 26 \mathbf { i } + 15 \mathbf { j } ) \mathrm { km }\). It is not safe for ships to be within 1.3 km of the lighthouse.
    1. Find the value of \(t\) when \(S\) is closest to the lighthouse.
    2. Hence determine whether it is safe for \(S\) to continue its course.
Edexcel M1 2024 January Q8
12 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e59a66b8-c2ad-41fd-9959-9d21e9455c37-24_346_961_246_543} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A fixed rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\)
A small smooth pulley is fixed at the top of the plane.
One end of a light inextensible string is attached to a particle \(P\) which is at rest on the plane. The string passes over the pulley and the other end of the string is attached to a particle \(Q\) which hangs vertically below the pulley, as shown in Figure 5. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(0.5 m\)
The string from \(P\) to the pulley lies along a line of greatest slope of the plane.
The coefficient of friction between \(P\) and the plane is \(\mu\).
The system is in limiting equilibrium with the string taut and \(P\) is on the point of slipping up the plane.
  1. Find the value of \(\mu\). The string breaks and \(P\) begins to move down the plane.
    When particle \(P\) has travelled a distance of 0.8 m down the plane, the speed of \(P\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. Find the value of \(V\).
Edexcel M1 2014 June Q1
5 marks Moderate -0.8
  1. Two small smooth balls \(A\) and \(B\) have mass 0.6 kg and 0.9 kg respectively. They are moving in a straight line towards each other in opposite directions on a smooth horizontal floor and collide directly. Immediately before the collision the speed of \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) immediately after the collision and \(B\) is brought to rest by the collision.
Find
  1. the value of \(v\),
  2. the magnitude of the impulse exerted on \(A\) by \(B\) in the collision.
Edexcel M1 2014 June Q2
8 marks Standard +0.3
2. A ball is thrown vertically upwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\), which is \(h\) metres above the ground. The ball moves freely under gravity until it hits the ground 5 s later.
  1. Find the value of \(h\). A second ball is thrown vertically downwards with speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) and moves freely under gravity until it hits the ground. The first ball hits the ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the second ball hits the ground with speed \(\frac { 3 } { 4 } V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the value of \(w\).
Edexcel M1 2014 June Q3
12 marks Standard +0.3
3. A particle \(P\) of mass 1.5 kg is placed at a point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.6
  1. Show that \(P\) rests in equilibrium at \(A\). A horizontal force of magnitude \(X\) newtons is now applied to \(P\), as shown in Figure 1. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{edcc4603-f006-4c4f-a4e5-063cab41da98-04_236_584_667_680} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The particle is on the point of moving up the plane.
  2. Find
    1. the magnitude of the normal reaction of the plane on \(P\),
    2. the value of \(X\).
Edexcel M1 2014 June Q4
10 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{edcc4603-f006-4c4f-a4e5-063cab41da98-06_262_1132_223_415} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A plank \(A B\), of length 6 m and mass 4 kg , rests in equilibrium horizontally on two supports at \(C\) and \(D\), where \(A C = 2 \mathrm {~m}\) and \(D B = 1 \mathrm {~m}\). A brick of mass 2 kg rests on the plank at \(A\) and a brick of mass 3 kg rests on the plank at \(B\), as shown in Figure 2. The plank is modelled as a uniform rod and all bricks are modelled as particles.
  1. Find the magnitude of the reaction exerted on the plank
    1. by the support at \(C\),
    2. by the support at \(D\). The 3 kg brick is now removed and replaced with a brick of mass \(x \mathrm {~kg}\) at \(B\). The plank remains horizontal and in equilibrium but the reactions on the plank at \(C\) and at \(D\) now have equal magnitude.
  2. Find the value of \(x\).