OCR M1 (Mechanics 1)

\includegraphics{figure_1} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\). A smooth ring \(R\) of mass \(m\) kg is threaded on the string and is pulled by a force of magnitude \(1.6\) N acting upwards at \(45°\) to the horizontal. The section \(AR\) of the string makes an angle of \(30°\) with the downward vertical and the section \(BR\) is horizontal (see diagram). The ring is in equilibrium with the string taut.

1. Give a reason why the tension in the part \(AR\) of the string is the same as that in the part \(BR\). [1]
2. Show that the tension in the string is \(0.754\) N, correct to 3 significant figures. [3]
3. Find the value of \(m\). [3]

\includegraphics{figure_1} Particles \(P\) and \(Q\), of masses \(0.3\) kg and \(0.4\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in motion with the string taut and with each of the particles moving vertically. The downward acceleration of \(P\) is \(a\) m s\(^{-2}\) (see diagram).

1. Show that \(a = -1.4\). [4]

Initially \(P\) and \(Q\) are at the same horizontal level. \(P\)'s initial velocity is vertically downwards and has magnitude \(2.8\) m s\(^{-1}\).

1. Assuming that \(P\) does not reach the floor and that \(Q\) does not reach the pulley, find the time taken for \(P\) to return to its initial position. [3]

Each of two wagons has an unloaded mass of \(1200\) kg. One of the wagons carries a load of mass \(m\) kg and the other wagon is unloaded. The wagons are moving towards each other on the same rails, each with speed \(3\) m s\(^{-1}\), when they collide. Immediately after the collision the loaded wagon is at rest and the speed of the unloaded wagon is \(5\) m s\(^{-1}\). Find the value of \(m\). [5]

A trailer of mass \(600\) kg is attached to a car of mass \(1100\) kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration \(0.8\) m s\(^{-2}\).

1. Given that the force exerted on the trailer by the tow-bar is \(700\) N, find the resistance to motion of the trailer. [4]
2. Given also that the driving force of the car is \(2100\) N, find the resistance to motion of the car. [3]

\includegraphics{figure_1} Two horizontal forces \(\mathbf{P}\) and \(\mathbf{Q}\) act at the origin \(O\) of rectangular coordinates \(Oxy\) (see diagram). The components of \(\mathbf{P}\) in the \(x\)- and \(y\)-directions are \(14\) N and \(5\) N respectively. The components of \(\mathbf{Q}\) in the \(x\)- and \(y\)-directions are \(-9\) N and \(7\) N respectively.

1. Write down the components, in the \(x\)- and \(y\)-directions, of the resultant of \(\mathbf{P}\) and \(\mathbf{Q}\). [2]
2. Hence find the magnitude of this resultant, and the angle the resultant makes with the positive \(x\)-axis. [4]

\includegraphics{figure_2} Particles \(A\) and \(B\), of masses \(0.2\) kg and \(0.3\) kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest at a fixed point and \(B\) hangs vertically below \(A\). Particle \(A\) is now released. As the particles fall the air resistance acting on \(A\) is \(0.4\) N and the air resistance acting on \(B\) is \(0.25\) N (see diagram). The downward acceleration of each of the particles is \(a\) m s\(^{-2}\) and the tension in the string is \(T\) N.

1. Write down two equations in \(a\) and \(T\) obtained by applying Newton's second law to \(A\) and to \(B\). [4]
2. Find the values of \(a\) and \(T\). [3]

\includegraphics{figure_2} An object of mass \(0.08\) kg is attached to one end of a light inextensible string. The other end of the string is attached to the underside of the roof inside a furniture van. The van is moving horizontally with constant acceleration \(1.25\) m s\(^{-2}\). The string makes a constant angle \(\alpha\) with the downward vertical and the tension in the string is \(T\) N (see diagram).

1. By applying Newton's second law horizontally to the object, find the value of \(T \sin \alpha\). [2]
2. Find the value of \(T\). [5]

\includegraphics{figure_2} Forces of magnitudes \(6.5\) N and \(2.5\) N act at a point in the directions shown. The resultant of the two forces has magnitude \(R\) N and acts at right angles to the force of magnitude \(2.5\) N (see diagram).

1. Show that \(\theta = 22.6°\), correct to 3 significant figures. [3]
2. Find the value of \(R\). [3]

\includegraphics{figure_2} Three horizontal forces of magnitudes \(15\) N, \(11\) N and \(13\) N act on a particle \(P\) in the directions shown in the diagram. The angles \(\alpha\) and \(\beta\) are such that \(\sin \alpha = 0.28\), \(\cos \alpha = 0.96\), \(\sin \beta = 0.8\) and \(\cos \beta = 0.6\).

1. Show that the component, in the \(y\)-direction, of the resultant of the three forces is zero. [4]
2. Find the magnitude of the resultant of the three forces. [3]
3. State the direction of the resultant of the three forces. [1]

\includegraphics{figure_2} A particle starts from the point \(A\) and travels in a straight line. The diagram shows the \((t, v)\) graph, consisting of three straight line segments, for the motion of the particle during the interval \(0 \leq t \leq 290\).

1. Find the value of \(t\) for which the distance of the particle from \(A\) is greatest. [2]
2. Find the displacement of the particle from \(A\) when \(t = 290\). [3]
3. Find the total distance travelled by the particle during the interval \(0 \leq t \leq 290\). [2]

Two small spheres \(P\) and \(Q\) have masses \(0.1\) kg and \(0.2\) kg respectively. The spheres are moving directly towards each other on a horizontal plane and collide. Immediately before the collision \(P\) has speed \(4\) m s\(^{-1}\) and \(Q\) has speed \(3\) m s\(^{-1}\). Immediately after the collision the spheres move away from each other, \(P\) with speed \(u\) m s\(^{-1}\) and \(Q\) with speed \((3.5 - u)\) m s\(^{-1}\).

1. Find the value of \(u\). [4]

After the collision the spheres both move with deceleration of magnitude \(5\) m s\(^{-2}\) until they come to rest on the plane.

1. Find the distance \(PQ\) when both \(P\) and \(Q\) are at rest. [4]

A motorcyclist starts from rest at a point \(O\) and travels in a straight line. His velocity after \(t\) seconds is \(v\) m s\(^{-1}\), for \(0 \leq t \leq T\), where \(v = 7.2t - 0.45t^2\). The motorcyclist's acceleration is zero when \(t = T\).

1. Find the value of \(T\). [4]
2. Show that \(v = 28.8\) when \(t = T\). [1]

For \(t \geq T\) the motorcyclist travels in the same direction as before, but with constant speed \(28.8\) m s\(^{-1}\).

1. Find the displacement of the motorcyclist from \(O\) when \(t = 31\). [6]

A man travels \(360\) m along a straight road. He walks for the first \(120\) m at \(1.5\) m s\(^{-1}\), runs the next \(180\) m at \(4.5\) m s\(^{-1}\), and then walks the final \(60\) m at \(1.5\) m s\(^{-1}\). The man's displacement from his starting point after \(t\) seconds is \(x\) metres.

1. Sketch the \((t, x)\) graph for the journey, showing the values of \(t\) for which \(x = 120, 300\) and \(360\). [5]

A woman jogs the same \(360\) m route at constant speed, starting at the same instant as the man and finishing at the same instant as the man.

1. Draw a dotted line on your \((t, x)\) graph to represent the woman's journey. [1]
2. Calculate the value of \(t\) at which the man overtakes the woman. [5]

\includegraphics{figure_3} A block \(B\) of mass \(0.4\) kg and a particle \(P\) of mass \(0.3\) kg are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(B\) is in contact with the table and the part of the string between \(B\) and the pulley is horizontal. \(P\) hangs freely below the pulley (see diagram).

1. The system is in limiting equilibrium with the string taut and \(P\) on the point of moving downwards. Find the coefficient of friction between \(B\) and the table. [5]
2. A horizontal force of magnitude \(X\) N, acting directly away from the pulley, is now applied to \(B\). The system is again in limiting equilibrium with the string taut, and with \(P\) now on the point of moving upwards. Find the value of \(X\). [3]

A particle moves downwards on a smooth plane inclined at an angle \(\alpha\) to the horizontal. The particle passes through the point \(P\) with speed \(u\) m s\(^{-1}\). The particle travels \(2\) m during the first \(0.8\) s after passing through \(P\), then a further \(6\) m in the next \(1.2\) s. Find

1. the value of \(u\) and the acceleration of the particle, [7]
2. the value of \(\alpha\) in degrees. [2]

\includegraphics{figure_4} A block of mass \(2\) kg is at rest on a rough horizontal plane, acted on by a force of magnitude \(12\) N at an angle of \(15°\) upwards from the horizontal (see diagram).

1. Find the frictional component of the contact force exerted on the block by the plane. [2]
2. Show that the normal component of the contact force exerted on the block by the plane has magnitude \(16.5\) N, correct to 3 significant figures. [2]

It is given that the block is on the point of sliding.

1. Find the coefficient of friction between the block and the plane. [2]

The force of magnitude \(12\) N is now replaced by a horizontal force of magnitude \(20\) N. The block starts to move.

1. Find the acceleration of the block. [5]

A cyclist travels along a straight road. Her velocity \(v\) m s\(^{-1}\), at time \(t\) seconds after starting from a point \(O\), is given by \(v = 2\) for \(0 \leq t \leq 10\), \(v = 0.03t^2 - 0.3t + 2\) for \(t \geq 10\).

1. Find the displacement of the cyclist from \(O\) when \(t = 10\). [1]
2. Show that, for \(t \geq 10\), the displacement of the cyclist from \(O\) is given by the expression \(0.01t^3 - 0.15t^2 + 2t + 5\). [4]
3. Find the time when the acceleration of the cyclist is \(0.6\) m s\(^{-2}\). Hence find the displacement of the cyclist from \(O\) when her acceleration is \(0.6\) m s\(^{-2}\). [5]

\includegraphics{figure_4} Three uniform spheres \(L\), \(M\) and \(N\) have masses \(0.8\) kg, \(0.6\) kg and \(0.7\) kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with \(M\) between \(L\) and \(N\). The sphere \(L\) is moving towards \(M\) with speed \(4\) m s\(^{-1}\) and the spheres \(M\) and \(N\) are moving towards \(L\) with speeds \(2\) m s\(^{-1}\) and \(0.5\) m s\(^{-1}\) respectively (see diagram).

1. \(L\) collides with \(M\). As a result of this collision the direction of motion of \(M\) is reversed, and its speed remains \(2\) m s\(^{-1}\). Find the speed of \(L\) after the collision. [4]
2. \(M\) then collides with \(N\).
   1. Find the total momentum of \(M\) and \(N\) in the direction of \(M\)'s motion before this collision takes place, and deduce that the direction of motion of \(N\) is reversed as a result of this collision. [4]
   2. Given that \(M\) is at rest immediately after this collision, find the speed of \(N\) immediately after this collision. [2]

\includegraphics{figure_5} Two small rings \(A\) and \(B\) are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. \(A\) is above \(B\). A horizontal force is applied to a point \(P\) of the string. Both parts \(AP\) and \(BP\) of the string are taut. The system is in equilibrium with angle \(BAP = \alpha\) and angle \(ABP = \beta\) (see diagram). The weight of \(A\) is \(2\) N and the tensions in the parts \(AP\) and \(BP\) of the string are \(7\) N and \(T\) N respectively. It is given that \(\cos \alpha = 0.28\) and \(\sin \alpha = 0.96\), and that \(A\) is in limiting equilibrium.

1. Find the coefficient of friction between the wire and the ring \(A\). [7]
2. By considering the forces acting at \(P\), show that \(T \cos \beta = 1.96\). [2]
3. Given that there is no frictional force acting on \(B\), find the mass of \(B\). [3]

A man drives a car on a horizontal straight road. At \(t = 0\), where the time \(t\) is in seconds, the car runs out of petrol. At this instant the car is moving at \(12\) m s\(^{-1}\). The car decelerates uniformly, coming to rest when \(t = 8\). The man then walks back along the road at \(0.7\) m s\(^{-1}\) until he reaches a petrol station a distance of \(420\) m from his car. After his arrival at the petrol station it takes him \(250\) s to obtain a can of petrol. He is then given a lift back to his car on a motorcycle. The motorcycle starts from rest and accelerates uniformly until its speed is \(20\) m s\(^{-1}\); it then decelerates uniformly, coming to rest at the stationary car at time \(t = T\).

1. Sketch the shape of the \((t, v)\) graph for the man for \(0 \leq t \leq T\). [Your sketch need not be drawn to scale; numerical values need not be shown.] [5]
2. Find the deceleration of the car for \(0 < t < 8\). [2]
3. Find the value of \(T\). [4]

A block of mass \(m\) kg is at rest on a horizontal plane. The coefficient of friction between the block and the plane is \(0.2\).

1. When a horizontal force of magnitude \(5\) N acts on the block, the block is on the point of slipping. Find the value of \(m\). [3]

1. \includegraphics{figure_5ii} When a force of magnitude \(P\) N acts downwards on the block at an angle \(\alpha\) to the horizontal, as shown in the diagram, the frictional force on the block has magnitude \(6\) N and the block is again on the point of slipping. Find
   1. the value of \(\alpha\) in degrees,
   2. the value of \(P\).
   [8]

A particle starts from rest at a point \(A\) at time \(t = 0\), where \(t\) is in seconds. The particle moves in a straight line. For \(0 \leq t \leq 4\) the acceleration is \(1.8t\) m s\(^{-2}\), and for \(4 \leq t \leq 7\) the particle has constant acceleration \(7.2\) m s\(^{-2}\).

1. Find an expression for the velocity of the particle in terms of \(t\), valid for \(0 \leq t \leq 4\). [3]
2. Show that the displacement of the particle from \(A\) is \(19.2\) m when \(t = 4\). [4]
3. Find the displacement of the particle from \(A\) when \(t = 7\). [5]

A particle of mass \(0.04\) kg is acted on by a force of magnitude \(P\) N in a direction at an angle \(\alpha\) to the upward vertical.

1. The resultant of the weight of the particle and the force applied to the particle acts horizontally. Given that \(\alpha = 20°\) find
   1. the value of \(P\), [3]
   2. the magnitude of the resultant, [2]
   3. the magnitude of the acceleration of the particle. [2]
2. It is given instead that \(P = 0.08\) and \(\alpha = 90°\). Find the magnitude and direction of the resultant force on the particle. [5]

\includegraphics{figure_6} A smooth ring \(R\) of weight \(W\) N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). A horizontal force of magnitude \(P\) N acts on \(R\). The system is in equilibrium with the string taut; \(AR\) makes an angle \(\alpha\) with the downward vertical and \(BR\) makes an angle \(\beta\) with the upward vertical (see Fig. 1).

1. By considering the vertical components of the forces acting on \(R\), show that \(\alpha < \beta\). [3]

1. \includegraphics{figure_6ii} It is given that when \(P = 14\), \(AR = 0.4\) m, \(BR = 0.3\) m and the distance of \(R\) from the vertical line \(AB\) is \(0.24\) m (see Fig. 2). Find
   1. the tension in the string, [3]
   2. the value of \(W\). [3]
2. For the case when \(P = 0\),
   1. describe the position of \(R\), [1]
   2. state the tension in the string. [1]

\includegraphics{figure_6} A train of total mass \(80000\) kg consists of an engine \(E\) and two trucks \(A\) and \(B\). The engine \(E\) and truck \(A\) are connected by a rigid coupling \(X\), and trucks \(A\) and \(B\) are connected by another rigid coupling \(Y\). The couplings are light and horizontal. The train is moving along a straight horizontal track. The resistances to motion acting on \(E\), \(A\) and \(B\) are \(10500\) N, \(3000\) N and \(1500\) N respectively (see diagram).

1. By modelling the whole train as a single particle, show that it is decelerating when the driving force of the engine is less than \(15000\) N. [2]
2. Show that, when the magnitude of the driving force is \(35000\) N, the acceleration of the train is \(0.25\) m s\(^{-2}\). [2]
3. Hence find the mass of \(E\), given that the tension in the coupling \(X\) is \(8500\) N when the magnitude of the driving force is \(35000\) N. [3]

The driving force is replaced by a braking force of magnitude \(15000\) N acting on the engine. The force exerted by the coupling \(Y\) is zero.

1. Find the mass of \(B\). [5]
2. Show that the coupling \(X\) exerts a forward force of magnitude \(1500\) N on the engine. [2]

\includegraphics{figure_6} The diagram shows the \((t, v)\) graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for \(25\) s, coming to rest \(8\) m above ground level.

1. Find the greatest speed reached by the hoist during this stage. [2]

The second stage consists of a \(40\) s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest \(40\) m above ground level, arriving \(135\) s after leaving ground level. The hoist accelerates at \(0.02\) m s\(^{-2}\) for the first \(40\) s of the third stage, reaching a speed of \(V\) m s\(^{-1}\). Find

1. the value of \(V\), [3]
2. the length of time during the third stage for which the hoist is moving at constant speed, [4]
3. the deceleration of the hoist in the final part of the third stage. [3]

\includegraphics{figure_7} A car \(P\) starts from rest and travels along a straight road for \(600\) s. The \((t, v)\) graph for the journey is shown in the diagram. This graph consists of three straight line segments. Find

1. the distance travelled by \(P\), [3]
2. the deceleration of \(P\) during the interval \(500 < t < 600\). [2]

Another car \(Q\) starts from rest at the same instant as \(P\) and travels in the same direction along the same road for \(600\) s. At time \(t\) s after starting the velocity of \(Q\) is \((600t^2 - t^3) \times 10^{-6}\) m s\(^{-1}\).

1. Find an expression in terms of \(t\) for the acceleration of \(Q\). [2]
2. Find how much less \(Q\)'s deceleration is than \(P\)'s when \(t = 550\). [2]
3. Show that \(Q\) has its maximum velocity when \(t = 400\). [2]
4. Find how much further \(Q\) has travelled than \(P\) when \(t = 400\). [6]

\includegraphics{figure_7} \(PQ\) is a line of greatest slope, of length \(4\) m, on a smooth plane inclined at \(30°\) to the horizontal. Particles \(A\) and \(B\), of masses \(0.15\) kg and \(0.5\) kg respectively, move along \(PQ\) with \(A\) below \(B\). The particles are both moving upwards, \(A\) with speed \(8\) m s\(^{-1}\) and \(B\) with speed \(2\) m s\(^{-1}\), when they collide at the mid-point of \(PQ\) (see diagram). Particle \(A\) is instantaneously at rest immediately after the collision.

1. Show that \(B\) does not reach \(Q\) in the subsequent motion. [8]
2. Find the time interval between the instant of \(A\)'s arrival at \(P\) and the instant of \(B\)'s arrival at \(P\). [6]

A particle of mass \(0.1\) kg is at rest at a point \(A\) on a rough plane inclined at \(15°\) to the horizontal. The particle is given an initial velocity of \(6\) m s\(^{-1}\) and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after \(1.5\) s.

1. Find the coefficient of friction between the particle and the plane. [7]
2. Show that, after coming to instantaneous rest, the particle moves down the plane. [2]
3. Find the speed with which the particle passes through \(A\) during its downward motion. [6]

A particle \(P\) of mass \(0.5\) kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of \(40°\) to the horizontal. \(P\) reaches its highest point and then moves back down the plane. The coefficient of friction between \(P\) and the plane is \(0.6\).

1. Show that the magnitude of the frictional force acting on \(P\) is \(2.25\) N, correct to 3 significant figures. [3]
2. Find the acceleration of \(P\) when it is moving
   1. up the plane,
   2. down the plane.
   [4]
3. When \(P\) is moving up the plane, it passes through a point \(A\) with speed \(4\) m s\(^{-1}\).
   1. Find the length of time before \(P\) reaches its highest point.
   2. Find the total length of time for \(P\) to travel from the point \(A\) to its highest point and back to \(A\).
   [8]