3.03d Newton's second law: 2D vectors

Two particles, \(P\) and \(Q\), are initially at rest at the same point on a horizontal plane. A force of \(\begin{bmatrix} 4 \\ 0 \end{bmatrix}\) N is applied to \(P\). A force of \(\begin{bmatrix} 8 \\ 15 \end{bmatrix}\) N is applied to \(Q\).

1. Calculate, to the nearest degree, the acute angle between the two forces. [2 marks]
2. The particles begin to move under the action of the respective forces. \(P\) and \(Q\) have the same mass. \(P\) has an acceleration of magnitude 5 m s\(^{-2}\) Find the magnitude of the acceleration of \(Q\). [3 marks]

\includegraphics{figure_17} A car and caravan, connected by a tow bar, move forward together along a horizontal road. Their velocity \(v\) m s\(^{-1}\) at time \(t\) seconds, for \(0 \leq t < 20\), is given by $$v = 0.5t + 0.01t^2$$

1. Show that when \(t = 15\) their acceleration is 0.8 m s\(^{-2}\) [2 marks]
2. The car has a mass of 1500 kg The caravan has a mass of 850 kg When \(t = 15\) the tension in the tow bar is 800 N and the car experiences a resistance force of 100 N
   1. Find the total resistance force experienced by the caravan when \(t = 15\) [2 marks]
   2. Find the driving force being applied by the car when \(t = 15\) [3 marks]
3. State one assumption you have made about the tow bar. [1 mark]

A resultant force of \(\begin{bmatrix} -2 \\ 6 \end{bmatrix}\) N acts on a particle. The acceleration of the particle is \(\begin{bmatrix} -6 \\ y \end{bmatrix} \text{ m s}^{-2}\) Find the value of \(y\) Circle your answer. [1 mark] \(2\) \quad \(3\) \quad \(10\) \quad \(18\)

A rescue van is towing a broken-down car by using a tow bar. The van and the car are moving with a constant acceleration of \(0.6 \text{ m s}^{-2}\) along a straight horizontal road as shown in the diagram below. \includegraphics{figure_18} The van has a total mass of 2780 kg The car has a total mass of 1620 kg The van experiences a driving force of \(D\) newtons. The van experiences a total resistance force of \(R\) newtons. The car experiences a total resistance force of \(0.6R\) newtons.

1. The tension in the tow bar, \(T\) newtons, may be modelled by $$T = kD - 18$$ where \(k\) is a constant. Find \(k\) [5 marks]
2. State one assumption that must be made in answering part (a). [1 mark]

A buggy is pulling a roller-skater, in a straight line along a horizontal road, by means of a connecting rope as shown in the diagram. \includegraphics{figure_6} The combined mass of the buggy and driver is 410 kg A driving force of 300 N and a total resistance force of 140 N act on the buggy. The mass of the roller-skater is 72 kg A total resistance force of R newtons acts on the roller-skater. The buggy and the roller-skater have an acceleration of 0.2 m s\(^{-2}\)

1. 1. Find R. [3 marks]
   2. Find the tension in the rope. [3 marks]
2. State a necessary assumption that you have made. [1 mark]
3. The roller-skater releases the rope at a point A, when she reaches a speed of 6 m s\(^{-1}\) She continues to move forward, experiencing the same resistance force. The driver notices a change in motion of the buggy, and brings it to rest at a distance of 20 m from A.
   1. Determine whether the roller-skater will stop before reaching the stationary buggy. Fully justify your answer. [5 marks]
   2. Explain the change in motion that the driver noticed. [2 marks]

Lizzie is sat securely on a wooden sledge. The combined mass of Lizzie and the sledge is \(M\) kilograms. The sledge is being pulled forward in a straight line along a horizontal surface by means of a light inextensible rope, which is attached to the front of the sledge. This rope stays inclined at an acute angle \(\theta\) above the horizontal and remains taut as the sledge moves forward. \includegraphics{figure_17} The sledge remains in contact with the surface throughout. The coefficient of friction between the sledge and the surface is \(\mu\) and there are no other resistance forces. Lizzie and the sledge move forward with constant acceleration, \(a \text{ m s}^{-2}\) The tension in the rope is a constant \(T\) Newtons.

1. Show that $$T = \frac{M(a + \mu g)}{\cos \theta + \mu \sin \theta}$$ [7 marks]
2. It is known that when \(M = 30\), \(\theta = 30°\), and \(T = 40\), the sledge remains at rest. Lizzie uses these values with the relationship formed in part (a) to find the value for \(\mu\) Explain why her value for \(\mu\) may be incorrect. [2 marks]

Two forces, \(\mathbf{F_1}\) and \(\mathbf{F_2}\), are acting on a particle of mass 3 kilograms. It is given that $$\mathbf{F_1} = \begin{pmatrix} a \\ 23 \end{pmatrix} \text{ newtons and } \mathbf{F_2} = \begin{pmatrix} 4 \\ b \end{pmatrix} \text{ newtons}$$ where \(a\) and \(b\) are constants. The particle has an acceleration of \(\begin{pmatrix} 4b \\ a \end{pmatrix}\) m s\(^{-2}\) Find the value of \(a\) and the value of \(b\) [4 marks]

Two heavy boxes, \(M\) and \(N\), are connected securely by a length of rope. The mass of \(M\) is 50 kilograms. The mass of \(N\) is 80 kilograms. \(M\) is placed near the bottom of a rough slope. The slope is inclined at 60° above the horizontal. The rope is passed over a smooth pulley at the top end of the slope so that \(N\) hangs with the rope vertical. The boxes are initially held in this position, with the rope taut and running parallel to the line of greatest slope, as shown in the diagram below. \includegraphics{figure_21} When the boxes are released, \(M\) moves up the slope as \(N\) descends, with acceleration \(a\) m s\(^{-2}\) The tension in the rope is \(T\) newtons.

1. Explain why the equation of motion for \(N\) is $$80g - T = 80a$$ [1 mark]
2. Show that the normal reaction force between \(M\) and the slope is \(25g\) newtons. [1 mark]
3. The coefficient of friction, \(\mu\), between the slope and \(M\) is such that \(0 \leq \mu \leq 1\) Show that $$a \geq \frac{(11 - 5\sqrt{3})g}{26}$$ [6 marks]
4. State one modelling assumption you have made throughout this question. [1 mark]

In this question use \(g = 9.8\) m s\(^{-2}\). The diagram shows a box, of mass 8.0 kg, being pulled by a string so that the box moves at a constant speed along a rough horizontal wooden board. The string is at an angle of 40° to the horizontal. The tension in the string is 50 newtons. \includegraphics{figure_16a} The coefficient of friction between the box and the board is \(\mu\) Model the box as a particle.

1. Show that \(\mu = 0.83\) [4 marks]
2. One end of the board is lifted up so that the board is now inclined at an angle of 5° to the horizontal. The box is pulled up the inclined board. The string remains at an angle of 40° to the board. The tension in the string is increased so that the box accelerates up the board at 3 m s\(^{-2}\) \includegraphics{figure_16b}
   1. Draw a diagram to show the forces acting on the box as it moves. [1 mark]
   2. Find the tension in the string as the box accelerates up the slope at 3 m s\(^{-2}\). [7 marks]

In this question use \(g = 9.8\,\text{m}\,\text{s}^{-2}\) A ride in a fairground consists of a hollow vertical cylinder of radius 4.6 metres with a horizontal floor. Stephi, who has mass 50 kilograms, stands inside the cylinder with her back against the curved surface. The cylinder begins to rotate about a vertical axis through the centre of the cylinder. When the cylinder is rotating at a constant angular speed of \(\omega\) radians per second, the magnitude of the normal reaction between Stephi and the curved surface is 980 newtons. The floor is lowered and Stephi remains against the curved surface with her feet above the floor, as shown in the diagram. \includegraphics{figure_4}

1. Explain, with the aid of a force diagram, why the magnitude of the frictional force acting on Stephi is 490 newtons. [2 marks]
2. Find \(\omega\) [3 marks]
3. State one modelling assumption that you have used in this question. Explain the effect of this assumption. [2 marks]

At time \(t\) seconds, where \(t \geq 0\), a particle P of mass 2 kg is moving on a smooth horizontal surface. The particle moves under the action of a constant horizontal force of \((-2\mathbf{i} + 6\mathbf{j})\) N and a variable horizontal force of \((2\cos 2t \mathbf{i} + 4\sin t \mathbf{j})\) N. The acceleration of P at time \(t\) seconds is \(\mathbf{a}\) m s\(^{-2}\).

1. Find \(\mathbf{a}\) in terms of \(t\). [2]

The particle P is at rest when \(t = 0\).

1. Determine the speed of P at the instant when \(t = 2\). [5]

\includegraphics{figure_12} Fig. 12 shows a hemispherical bowl. The rim of this bowl is a circle with centre O and radius \(r\). The bowl is fixed with its rim horizontal and uppermost. A particle P, of mass \(m\), is connected by a light inextensible string of length \(l\) to the lowest point A on the bowl and describes a horizontal circle with constant angular speed \(\omega\) on the smooth inner surface of the bowl. The string is taut, and AP makes an angle \(\alpha\) with the vertical.

1. Show that the normal contact force between P and the bowl is of magnitude \(mg + 2mr\omega^2\cos^2\alpha\). [9]
2. Deduce that \(g < r\omega^2(k_1 + k_2\cos^2\alpha)\), stating the value of the constants \(k_1\) and \(k_2\). [3]

Fig. 5 shows a light inextensible string of length 3.3 m passing through a small smooth ring R. The ends of the string are attached to fixed points A and B, where A is vertically above B. The ring R has mass 0.27 kg and is moving with constant speed in a horizontal circle of radius 1.2 m. The distances AR and BR are 2 m and 1.3 m respectively. \includegraphics{figure_5}

1. Show that the tension in the string is 6.37 N. [4]
2. Find the speed of R. [4]

The diagram shows two objects \(A\) and \(B\), of mass 3 kg and 5 kg respectively, connected by a light inextensible string passing over a light smooth pulley fixed at the end of a smooth horizontal surface. Object \(A\) lies on the horizontal surface and object \(B\) hangs freely below the pulley. \includegraphics{figure_8} Initially, \(B\) is supported so that the objects are at rest with the string just taut. Object \(B\) is then released.

1. Find the magnitude of the acceleration of \(A\) and the tension in the string. [6]
2. State briefly what effect a rough pulley would have on the tension in the string. [1]

A person, of mass 68 kg, stands in a lift which is moving upwards with constant acceleration. The lift is of mass 770 kg and the tension in the lift cable is 8000 N.

1. Determine the acceleration of the lift, giving your answer correct to two decimal places. [3]
2. State whether the lift is getting faster, staying at the same speed or slowing down. [1]
3. Calculate the magnitude of the reaction of the floor of the lift on the person. [3]

The diagram below shows a forklift truck being used to raise two boxes, \(P\) and \(Q\), vertically. Box \(Q\) rests on horizontal forks and box \(P\) rests on top of box \(Q\). Box \(P\) has mass 25 kg and box \(Q\) has mass 55 kg. \includegraphics{figure_7}

1. When the boxes are moving upwards with uniform acceleration, the reaction of the horizontal forks on box \(Q\) is 820 N. Calculate the magnitude of the acceleration. [3]
2. Calculate the reaction of box \(Q\) on box \(P\) when they are moving vertically upwards with constant speed. [1]

A particle, of mass 4 kg, moves in a straight line under the action of a single force \(F\) N, whose magnitude at time \(t\) seconds is given by $$F = 12\sqrt{t} - 32 \quad \text{for} \quad t \geqslant 0.$$

1. Find the acceleration of the particle when \(t = 9\). [2]
2. Given that the particle has velocity \(-1\text{ms}^{-1}\) when \(t = 4\), find an expression for the velocity of the particle at \(t\) s. [3]
3. Determine whether the speed of the particle is increasing or decreasing when \(t = 9\). [2]

The diagram below shows an object \(A\), of mass \(2m\) kg, lying on a horizontal table. It is connected to another object \(B\), of mass \(m\) kg, by a light inextensible string, which passes over a smooth pulley \(P\), fixed at the edge of the table. Initially, object \(A\) is held at rest so that object \(B\) hangs freely with the string taut. \includegraphics{figure_9} Object \(A\) is then released.

1. When object \(B\) has moved downwards a vertical distance of 0·4 m, its speed is 1·2 ms\(^{-1}\). Use a formula for motion in a straight line with constant acceleration to show that the magnitude of the acceleration of \(B\) is 1·8 ms\(^{-2}\). [2]
2. During the motion, object \(A\) experiences a constant resistive force of 22 N. Find the value of \(m\) and hence determine the tension in the string. [6]
3. What assumption did the word 'inextensible' in the description of the string enable you to make in your solution? [1]

Two forces \(\mathbf{F}\) and \(\mathbf{G}\) acting on an object are such that $$\mathbf{F} = \mathbf{i} - 8\mathbf{j},$$ $$\mathbf{G} = 3\mathbf{i} + 11\mathbf{j}.$$ The object has a mass of 3 kg. Calculate the magnitude and direction of the acceleration of the object. [7]

An object of mass \(0 \cdot 5\) kg is thrown vertically upwards with initial speed \(24\) ms\(^{-1}\). The velocity of the object at time \(t\) seconds is \(v\) ms\(^{-1}\). During the upward motion, the object experiences a resistance to motion \(RN\), where \(R\) is proportional to \(v\). When the velocity of the object is \(0 \cdot 2\) ms\(^{-1}\) the resistance to motion is \(0 \cdot 08\) N.

1. Show that the upward motion of the object satisfies the differential equation $$\frac{\mathrm{d}v}{\mathrm{d}t} = -9 \cdot 8 - 0 \cdot 8\,v.$$ [3]
2. Find an expression for \(v\) at time \(t\). [6]
3. Determine the value of \(t\) when the object is at the highest point of the motion. [2]

A particle of mass 2 kg moves under the action of a constant force F N, where F is given by $$\mathbf{F} = -3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the magnitude of the acceleration of the particle. [3]
2. Given that at time \(t = 0\) seconds, the position vector of the particle is \(2\mathbf{i} - 7\mathbf{j} + 9\mathbf{k}\) and it is moving with velocity \(3\mathbf{i} - 2\mathbf{j} + \mathbf{k}\), find the position vector of the particle when \(t = 2\) seconds. [3]

A particle \(P\) of mass \(0.5\) kg moves on a horizontal plane such that its velocity vector \(\mathbf{v}\) ms\(^{-1}\) at time \(t\) seconds is given by $$\mathbf{v} = 12\cos(3t)\mathbf{i} - 5\sin(2t)\mathbf{j}.$$

1. Find an expression for the force acting on \(P\) at time \(t\) s. [3]
2. Given that when \(t = 0\), \(P\) has position vector \((\mathbf{4i} + \mathbf{7j})\) m relative to the origin \(O\), find an expression for the position vector of \(P\) at time \(t\) s. [4]
3. Hence determine the distance of \(P\) from \(O\) at time \(t = \frac{\pi}{2}\). [2]

The position vector \(\mathbf{x}\) metres at time \(t\) seconds of an object of mass 3 kg may be modelled by $$\mathbf{x} = 3\sin t \mathbf{i} - 4\cos 2t \mathbf{j} + 5\sin t \mathbf{k}.$$

1. Find an expression for the velocity vector \(\mathbf{v}\text{ ms}^{-1}\) at time \(t\) seconds and determine the least value of \(t\) when the object is instantaneously at rest. [7]
2. Write down the momentum vector at time \(t\) seconds. [1]
3. Find, in vector form, an expression for the force acting on the object at time \(t\) seconds. [3]

\includegraphics{figure_4} As shown in the diagram, \(AB\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length \(1\) m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m_1\) kg. One end of another light inextensible string of length \(1\) m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m_2\) kg, which is free to move on \(AB\). Initially, \(P\) moves in a horizontal circle of radius \(0.6\) m with constant angular velocity \(\omega\) rad s\(^{-1}\). The magnitude of the tension in string \(AP\) is denoted by \(T_1\) N while that in string \(PR\) is denoted by \(T_2\) N.

1. By considering forces on \(R\), express \(T_2\) in terms of \(m_2\). [2]
2. Show that
   1. \(T_1 = \frac{4g}{5}(m_1 + m_2)\). [2]
   2. \(\omega^2 = \frac{4g(m_1 + 2m_2)}{4m_1}\). [3]
3. Deduce that, in the case where \(m_1\) is much bigger than \(m_2\), \(\omega \approx 3.5\). [2]

In a different case, where \(m_1 = 2.5\) and \(m_2 = 2.8\), \(P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.

1. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega\) rad s\(^{-1}\) to zero. [5]

A car of mass \(600\)kg pulls a trailer of mass \(150\)kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude \(200\)N. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car is modelled as a force of magnitude \((200 + \lambda v)\)N, where \(\lambda\) is a constant. When the engine of the car is working at a constant rate of \(15\)kW, the car is moving at a constant speed of \(25\text{ms}^{-1}\).

1. Show that \(\lambda = 8\). [4]
2. Later on, the car is pulling the trailer up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = \frac{1}{15}\). The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude \(200\)N at all times. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude \((200 + 8v)\)N. The engine of the car is again working at a constant rate of \(15\)kW. When \(v = 10\), the towbar breaks. The trailer comes to instantaneous rest after moving a distance \(d\) metres up the road from the point where the towbar broke. Find the acceleration of the car immediately after the towbar breaks. [4]
3. Use the work-energy principle to find the value of \(d\). [4]