3.03d Newton's second law: 2D vectors

5. \begin{figure}[h] \end{figure} Two particles, \(P\) and \(Q\), have masses 3 kg and 5 kg respectively. The particles are connected by a light inextensible string which passes over a small smooth fixed pulley. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in Figure 3. Immediately after the particles are released from rest, \(P\) moves upwards with acceleration \(a \mathrm {~ms} ^ { - 2 }\) and the tension in the string is \(T\) newtons.

1. Write down an equation of motion for \(P\).
2. Find the value of \(T\). The total force acting on the pulley due to the string has magnitude \(F\) newtons.
3. Find the value of \(F\). Initially, \(Q\) is 10 m above horizontal ground and \(P\) is more than 2 m below the pulley.
   At the instant when \(Q\) has descended a distance of 2 m , the string breaks and \(Q\) falls to the ground.
4. Find the speed of \(Q\) at the instant it hits the ground.

6 \includegraphics[max width=\textwidth, alt={}, center]{9ddae838-2639-4952-bbc0-3944a81e5762-3_401_1224_1315_456} The diagram shows a barge being towed along a canal by a force of 240 N at an angle of \(25 ^ { \circ }\) to its direction of motion. A force, \(F \mathrm {~N}\), perpendicular to the direction of motion, is applied to the barge to keep it moving in the direction shown.

1. Find the magnitude of \(F\).
2. The mass of the barge is 1100 kg and there is a resistance force of 100 N parallel to the direction of motion. Find the acceleration of the barge.

8 Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_213_1095_429_479} The acceleration of the system is \(0.3 \mathrm {~ms} ^ { - 2 }\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). Truck \(S\) is now subjected to an extra resistive force of 1800 N . The pulling force, \(P\), does not change.
2. Calculate the new acceleration of the trucks.
3. Calculate the force in the coupling between the trucks.

8 Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-5_215_1095_427_479} The acceleration of the system is \(0.3 \mathrm {~ms} ^ { - 2 }\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). Truck \(S\) is now subjected to an extra resistive force of 1800 N . The pulling force, \(P\), does not change.
2. Calculate the new acceleration of the trucks.
3. Calculate the force in the coupling between the trucks.

An aircraft moves along a straight horizontal runway with a constant acceleration of \(1.5 \mathrm {~ms} ^ { - 2 }\). Points \(A\) and \(B\) lie on the runway. The aircraft passes \(A\) with speed \(4 \mathrm {~ms} ^ { - 1 }\) and its speed at \(B\) must be at least \(78 \mathrm {~ms} ^ { - 1 }\) if it is to take off successfully. a) Find the speed of the aircraft 8 seconds after it passes \(A\).
b) Determine the minimum value of the distance \(A B\) for the aircraft to take off successfully. The diagram below shows an object \(A\), of mass 15 kg , lying on a smooth horizontal surface. It is connected to a box \(B\) by a light inextensible string which passes over a smooth pulley \(P\), fixed at the edge of the surface, so that box \(B\) hangs freely. An object \(C\) lies on the horizontal floor of box \(B\) so that the combined mass of \(B\) and \(C\) is 10 kg . \includegraphics[max width=\textwidth, alt={}, center]{77c62e6d-58e4-42d3-9982-5a8325e8e826-09_661_862_614_598} Initially, the system is held at rest with the string just taut. A horizontal force of magnitude 150 N is then applied to \(A\) in the direction \(P A\) so that box \(B\) is raised.
a) Find the magnitude of the acceleration of \(A\) and the tension in the string.
b) Given that object \(C\) has mass 4 kg , calculate the reaction of the floor of the box on object \(C\).
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1 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Sarah is going for a walk. She leaves her house and walks directly to the shop. She then walks directly from the shop to the park. Relative to her house:

• the shop has position vector \(\left( - \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { km }\),
• the park is 2 km away on a bearing of \(060 ^ { \circ }\).
  a) Show that the position vector of the park relative to the house is \(( \sqrt { 3 } \mathbf { i } + \mathbf { j } ) \mathrm { km }\).
  b) Determine the total distance walked by Sarah from her house to the park.
  c) By considering a modelling assumption you have made, explain why the answer you found in part (b) may not be the actual distance that Sarah walked.

Two particles \(P\) and \(Q\), of masses 0.5 kg and 0.3 kg respectively, are connected by a light inextensible string. The string is taut and \(P\) is vertically above \(Q\). A force of magnitude 10 N is applied to \(P\) vertically upwards. Find the acceleration of the particles and the tension in the string connecting them. [5]

A railway engine of mass 120000 kg is towing a coach of mass 60000 kg up a straight track inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.02\). There is a light rigid coupling, parallel to the track, connecting the engine and coach. The driving force produced by the engine is 125000 N and there are constant resistances to motion of 22000 N on the engine and 13000 N on the coach.

1. Find the acceleration of the engine and find the tension in the coupling. [5]

At an instant when the engine is travelling at 30 m s\(^{-1}\), it comes to a section of track inclined upwards at an angle \(\beta\) to the horizontal. The power produced by the engine is now 4500000 W and, as a result, the engine maintains a constant speed.

1. Assuming that the resistance forces remain unchanged, find the value of \(\beta\). [4]

A car of mass \(1500\) kg is pulling a trailer of mass \(300\) kg along a straight horizontal road at a constant speed of \(20\) m s\(^{-1}\). The system of the car and trailer is modelled as two particles, connected by a light rigid horizontal rod. The power of the car's engine is \(6000\) W. There are constant resistances to motion of \(R\) N on the car and \(80\) N on the trailer.

1. Find the value of \(R\). [2]
2. The power of the car's engine is increased to \(12\,500\) W. The resistance forces do not change. Find the acceleration of the car and trailer and the tension in the rod at an instant when the speed of the car is \(25\) m s\(^{-1}\). [5]

A car of mass 900 kg travels along a horizontal straight road with its engine working at a constant rate of \(P\) kW. The resistance to motion of the car is 550 N. Given that the acceleration of the car is \(0.2 \text{ m s}^{-2}\) at an instant when its speed is \(30 \text{ m s}^{-1}\), find the value of \(P\). [4]

A car of mass \(1200\) kg is driving along a straight horizontal road at a constant speed of \(15\) m s\(^{-1}\). There is a constant resistance to motion of \(350\) N.

1. Find the power of the car's engine. [1]

The car comes to a hill inclined at \(1°\) to the horizontal, still travelling at \(15\) m s\(^{-1}\).

1. The car starts to descend the hill with reduced power and with an acceleration of \(0.12\) m s\(^{-2}\). Given that there is no change in the resistance force, find the new power of the car's engine at the instant when it starts to descend the hill. [3]
2. When the car is travelling at \(20\) m s\(^{-1}\) down the hill, the power is cut off and the car gradually slows down. Assuming that the resistance force remains \(350\) N, find the distance travelled from the moment when the power is cut off until the speed of the car is reduced to \(18\) m s\(^{-1}\). [4]

A particle \(P\) of mass \(0.5\) kg moves in a straight line. At time \(t\) s the velocity of \(P\) is \(v\) m s\(^{-1}\) and its displacement from a fixed point \(O\) on the line is \(x\) m. The only forces acting on \(P\) are a force of magnitude \(\frac{150}{(x+1)^2}\) N in the direction of increasing displacement and a resistive force of magnitude \(\frac{450}{(x+1)^3}\) N. When \(t = 0\), \(x = 0\) and \(v = 20\). Find \(v\) in terms of \(x\), giving your answer in the form \(v = \frac{Ax + B}{(x + 1)}\), where \(A\) and \(B\) are constants to be determined. [6]

A particle \(P\) of mass \(2\text{kg}\) moving on a horizontal straight line has displacement \(x\text{m}\) from a fixed point \(O\) on the line and velocity \(v\text{ms}^{-1}\) at time \(t\). The only horizontal force acting on \(P\) is a variable force \(F\text{N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).

1. Find an expression for \(x\) in terms of \(t\). [4]

\includegraphics{figure_2} A particle \(P\) of mass 2 kg is pushed up a line of greatest slope of a rough plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 2. The force acts in the vertical plane which contains \(P\) and a line of greatest slope of the plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The coefficient of friction between \(P\) and the plane is 0.5 Given that the acceleration of \(P\) is 1.45 m s\(^{-2}\), find the value of \(X\). [10]

Two forces, \(\mathbf{F}_1\) and \(\mathbf{F}_2\), act on a particle \(A\). \(\mathbf{F}_1 = (2i - 3j)\) N and \(\mathbf{F}_2 = (pi + qj)\) N, where \(p\) and \(q\) are constants. Given that the resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is parallel to \((\mathbf{i} + 2\mathbf{j})\),

1. show that \(2p - q + 7 = 0\) [5] Given that \(q = 11\) and that the mass of \(A\) is 2 kg, and that \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only forces acting on \(A\),
2. find the magnitude of the acceleration of \(A\). [5]

[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors.] A particle \(A\) of mass 0.5 kg is at rest on a smooth horizontal plane. At time \(t = 0\), two forces, \(\mathbf{F}_1 = (-3\mathbf{i} + 2\mathbf{j})\) N and \(\mathbf{F}_2 = (p\mathbf{i} + q\mathbf{j})\) N, where \(p\) and \(q\) are constants, are applied to \(A\). Given that \(A\) moves in the direction of the vector \((\mathbf{i} - 2\mathbf{j})\),

1. show that \(2p + q - 4 = 0\) [4] Given that \(p = 5\)
2. Find the speed of \(A\) at time \(t = 4\) seconds. [5]

A particle \(P\), of mass 3 kg, moves under the action of two constant forces (6\(\mathbf{i}\) + 2\(\mathbf{j}\)) N and (3\(\mathbf{i}\) - 5\(\mathbf{j}\)) N.

1. Find, in the form (\(a\mathbf{i}\) + \(b\mathbf{j}\)) N, the resultant force \(\mathbf{F}\) acting on \(P\). [1]
2. Find, in degrees to one decimal place, the angle between \(\mathbf{F}\) and \(\mathbf{j}\). [3]
3. Find the acceleration of \(P\), giving your answer as a vector. [2]

The initial velocity of \(P\) is (-2\(\mathbf{i}\) + \(\mathbf{j}\)) m s\(^{-1}\).

1. Find, to 3 significant figures, the speed of \(P\) after 2 s. [5]

A particle \(P\) of mass 0.4 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. Initially the velocity of \(P\) is \((6\mathbf{i} - 2\mathbf{j})\) m s\(^{-1}\) and 4 s later the velocity of \(P\) is \((-14\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\).

1. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the acceleration of \(P\). [3]
2. Calculate the magnitude of \(\mathbf{F}\). [3]

A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. When \(t = 0\), \(P\) has velocity \((3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and at time \(t = 4\) s, \(P\) has velocity \((15\mathbf{i} - 4\mathbf{j})\) m s\(^{-1}\). Find

1. the acceleration of \(P\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\), [2]
2. the magnitude of \(\mathbf{F}\), [4]
3. the velocity of \(P\) at time \(t = 6\) s. [3]

\includegraphics{figure_4} Two particles \(A\) and \(B\) have masses \(5m\) and \(km\) respectively, where \(k < 5\). The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut, the hanging parts of the string vertical and with \(A\) and \(B\) at the same height above a horizontal plane, as shown in Figure 4. The system is released from rest. After release, \(A\) descends with acceleration \(\frac{1}{4}g\).

1. Show that the tension in the string as \(A\) descends is \(\frac{15}{4}mg\). [3]
2. Find the value of \(k\). [3]
3. State how you have used the information that the pulley is smooth. [1]

After descending for 1.2 s, the particle \(A\) reaches the plane. It is immediately brought to rest by the impact with the plane. The initial distance between \(B\) and the pulley is such that, in the subsequent motion, \(B\) does not reach the pulley.

1. Find the greatest height reached by \(B\) above the plane. [7]

A particle \(P\) of mass \(2 \text{ kg}\) moves in a plane under the action of a single constant force \(\mathbf{F}\) newtons. At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v} \text{ m s}^{-1}\). When \(t = 0\), \(\mathbf{v} = (-5\mathbf{i} + 7\mathbf{j})\) and when \(t = 3\), \(\mathbf{v} = (\mathbf{i} - 2\mathbf{j})\).

1. Find in degrees the angle between the direction of motion of \(P\) when \(t = 3\) and the vector \(\mathbf{j}\). [3]
2. Find the acceleration of \(P\). [2]
3. Find the magnitude of \(\mathbf{F}\). [3]
4. Find in terms of \(t\) the velocity of \(P\). [2]
5. Find the time at which \(P\) is moving parallel to the vector \(\mathbf{i} + \mathbf{j}\). [3]

A man travels in a lift to the top of a tall office block. The lift starts from rest on the ground floor and moves vertically. It comes to rest again at the top floor, having moved a vertical distance of \(27 \text{ m}\). The lift initially accelerates with a constant acceleration of \(2 \text{ m s}^{-1}\) until it reaches a speed of \(3 \text{ m s}^{-1}\). It then moves with a constant speed of \(3 \text{ m s}^{-1}\) for \(T\) seconds. Finally it decelerates with a constant deceleration for \(2.5 \text{ s}\) before coming to rest at the top floor.

1. Sketch a speed-time graph for the motion of the lift. [2]
2. Hence, or otherwise, find the value of \(T\). [3]
3. Sketch an acceleration-time graph for the motion of the lift. [3]

The mass of the man is \(80 \text{ kg}\) and the mass of the lift is \(120 \text{ kg}\). The lift is pulled up by means of a vertical cable attached to the top of the lift. By modelling the cable as light and inextensible, find

1. the tension in the cable when the lift is accelerating, [3]
2. the magnitude of the force exerted by the lift on the man during the last \(2.5 \text{ s}\) of the motion. [3]

\includegraphics{figure_4} Two particles \(P\) and \(Q\), of mass \(4\) kg and \(6\) kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. The coefficient of friction between each particle and the plane is \(\frac{2}{5}\). A constant force of magnitude \(40\) N is then applied to \(Q\) in the direction \(PQ\), as shown in Fig. 4.

1. Show that the acceleration of \(Q\) is \(1.2\) m s\(^{-2}\). [4]
2. Calculate the tension in the string when the system is moving. [3]
3. State how you have used the information that the string is inextensible. [1]

After the particles have been moving for \(7\) s, the string breaks. The particle \(Q\) remains under the action of the force of magnitude \(40\) N.

1. Show that \(P\) continues to move for a further \(3\) seconds. [5]
2. Calculate the speed of \(Q\) at the instant when \(P\) comes to rest. [4]

\includegraphics{figure_4} Figure 4 shows a lorry of mass 1600 kg towing a car of mass 900 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is at an angle of \(15°\) to the road. The lorry and the car experience constant resistances to motion of magnitude 600 N and 300 N respectively. The lorry's engine produces a constant horizontal force on the lorry of magnitude 1500 N. Find

1. the acceleration of the lorry and the car, [3]
2. the tension in the towbar. [4]

When the speed of the vehicles is \(6 \text{ m s}^{-1}\), the towbar breaks. Assuming that the resistance to the motion of the car remains of constant magnitude 300 N,

1. find the distance moved by the car from the moment the towbar breaks to the moment when the car comes to rest. [4]
2. State whether, when the towbar breaks, the normal reaction of the road on the car is increased, decreased or remains constant. Give a reason for your answer. [2]

A car of mass 800 kg pulls a trailer of mass 200 kg along a straight horizontal road using a light towbar which is parallel to the road. The horizontal resistances to motion of the car and the trailer have magnitudes 400 N and 200 N respectively. The engine of the car produces a constant horizontal driving force on the car of magnitude 1200 N. Find

1. the acceleration of the car and trailer, [3]
2. the magnitude of the tension in the towbar. [3]

The car is moving along the road when the driver sees a hazard ahead. He reduces the force produced by the engine to zero and applies the brakes. The brakes produce a force on the car of magnitude \(F\) newtons and the car and trailer decelerate. Given that the resistances to motion are unchanged and the magnitude of the thrust in the towbar is 100 N,

1. find the value of \(F\). [7]