3.03d Newton's second law: 2D vectors

A particle \(P\) of mass \(0.2\) kg oscillates with simple harmonic motion between the points \(A\) and \(B\), coming to rest at both points. The distance \(AB\) is \(0.2\) m, and \(P\) completes \(5\) oscillations every second.

1. Find, to \(3\) significant figures, the maximum resultant force exerted on \(P\). [6]

When the particle is at \(A\), it is struck a blow in the direction \(BA\). The particle now oscillates with simple harmonic motion with the same frequency as previously but twice the amplitude.

1. Find, to \(3\) significant figures, the speed of the particle immediately after it has been struck. [5]

A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the earth. The earth is modelled as a fixed sphere of radius \(R\). When \(S\) is at a distance \(x\) from the centre of the earth, the force exerted by the earth on \(S\) is directed towards the centre of the earth and has magnitude \(\frac{k}{x^2}\), where \(k\) is a constant.

1. Show that \(k = mgR^2\). [2]

Given that \(S\) starts from rest when its distance from the centre of the earth is \(2R\), and that air resistance can be ignored,

1. find the speed of \(S\) as it crashes into the surface of the earth. [7]

\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]

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_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]

Three forces act on a particle. The first force has magnitude \(P\text{ N}\) and acts horizontally due east. The second force has magnitude \(5\text{ N}\) and acts horizontally due west. The third force has magnitude \(2P\text{ N}\) and acts vertically upwards. The resultant of these three forces has magnitude \(25\text{ N}\).

1. Calculate \(P\) and the angle between the resultant and the vertical. [7]

The particle has mass \(3\text{ kg}\) and rests on a rough horizontal table. The coefficient of friction between the particle and the table is \(0.15\).

1. Find the acceleration of the particle, and state the direction in which it moves. [5]

The force acting on a particle of mass 1.5 kg is given by the vector \(\begin{pmatrix} 6 \\ 9 \end{pmatrix}\) N.

1. Give the acceleration of the particle as a vector. [2]
2. Calculate the angle that the acceleration makes with the direction \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\). [2]
3. At a certain point of its motion, the particle has a velocity of \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\) m s\(^{-1}\). Calculate the displacement of the particle over the next two seconds. [3]

A constant force, \(\mathbf{F}\), acts on a particle, \(P\), of mass 5 kg causing its velocity to change from \((-2\mathbf{i} + \mathbf{j})\) m s\(^{-1}\) to \((4\mathbf{i} - 7\mathbf{j})\) m s\(^{-1}\) in 2 seconds.

1. Find, in the form \(a\mathbf{i} + b\mathbf{j}\), the acceleration of \(P\). [2 marks]
2. Show that the magnitude of \(\mathbf{F}\) is 25 N and find, to the nearest degree, the acute angle between the line of action of \(\mathbf{F}\) and the vector \(\mathbf{j}\). [5 marks]

A rock of mass 8 kg is acted on by just the two forces \(-80\)k N and \((-\mathbf{i} + 16\mathbf{j} + 72\)k\()\) N, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane and k is a unit vector vertically upward.

1. Show that the acceleration of the rock is \(\left(\frac{1}{8}\mathbf{i} + 2\mathbf{j}\right)\) k\()\) ms\(^{-2}\). [2]

The rock passes through the origin of position vectors, O, with velocity \((\mathbf{i} - 4\mathbf{j} + 3\)k\()\) m s\(^{-1}\) and 4 seconds later passes through the point A.

1. Find the position vector of A. [3]
2. Find the distance OA. [1]
3. Find the angle that OA makes with the horizontal. [2]

A particle of mass 5 kg has constant acceleration. Initially, the particle is at \(\begin{pmatrix} -1 \\ 2 \end{pmatrix}\) m with velocity \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\) ms\(^{-1}\); after 4 seconds the particle has velocity \(\begin{pmatrix} 12 \\ 9 \end{pmatrix}\) ms\(^{-1}\).

1. Calculate the acceleration of the particle. [2]
2. Calculate the position of the particle at the end of the 4 seconds. [3]
3. Calculate the force acting on the particle. [2]

A particle has mass 6 kg. A single force \((24e^{-2t}\mathbf{i} - 12t^3\mathbf{j})\) newtons acts on the particle at time \(t\) seconds. No other forces act on the particle.

1. Find the acceleration of the particle at time \(t\). [2 marks]
2. At time \(t = 0\), the velocity of the particle is \((-7\mathbf{i} - 4\mathbf{j}) \text{ m s}^{-1}\). Find the velocity of the particle at time \(t\). [4 marks]
3. Find the speed of the particle when \(t = 0.5\). [4 marks]

A particle moves in a horizontal plane under the action of a single force, \(\mathbf{F}\) newtons. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed east and north respectively. At time \(t\) seconds, the velocity of the particle, \(\mathbf{v} \text{ m s}^{-1}\), is given by $$\mathbf{v} = (8t - t^4)\mathbf{i} + 6e^{-3t}\mathbf{j}$$

1. Find an expression for the acceleration of the particle at time \(t\). [2 marks]
2. The mass of the particle is \(2\) kg.
   1. Find an expression for the force \(\mathbf{F}\) acting on the particle at time \(t\). [2 marks]
   2. Find the magnitude of \(\mathbf{F}\) when \(t = 1\). [3 marks]
3. Find the value of \(t\) when \(\mathbf{F}\) acts due south. [2 marks]
4. When \(t = 0\), the particle is at the point with position vector \((3\mathbf{i} - 5\mathbf{j})\) metres. Find an expression for the position vector, \(\mathbf{r}\) metres, of the particle at time \(t\). [4 marks]

A particle \(P\) of mass 3 kg has position vector \(\mathbf{r} = (2t^2 - 4t)\mathbf{i} + (1 - t^2)\mathbf{j}\) m at time \(t\) seconds.

1. Find the velocity vector of \(P\) when \(t = 3\). [3 marks]
2. Find the magnitude of the force acting on \(P\), showing that this force is constant. [4 marks]

A car, of mass 1100 kg, pulls a trailer of mass 550 kg along a straight horizontal road by means of a rigid tow-bar. The car is accelerating at 1.2 ms\(^{-2}\) and the resistances to the motion of the car and trailer have magnitudes 500 N and 200 N respectively.

1. Show that the driving force produced by the engine of the car is 2680 N. [3 marks]
2. Find the tension in the tow-bar between the car and the trailer. [3 marks]
3. Find the rate, in kW, at which the car's engine is working when the car is moving with speed 18 ms\(^{-1}\). [2 marks]

When the car is moving at 18 ms\(^{-1}\) it starts to climb a straight hill which is inclined at \(6°\) to the horizontal. If the car's engine continues to work at the same rate and the resistances to motion remain the same as previously,

1. find the acceleration of the car at the instant when it starts to climb the hill. [3 marks]
2. Show that tension in the tow-bar remains unchanged. [3 marks]

At time \(t = 0\), a particle \(P\) of mass \(3\) kg is at rest at the point \(A\) with position vector \((j - 3k)\) m. Two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) then act on the particle \(P\) and it passes through the point \(B\) with position vector \((8i - 3j + 5k)\) m. Given that \(\mathbf{F}_1 = (4i - 2j + 5k)\) N and \(\mathbf{F}_2 = (8i - 4j + 7k)\) N and that \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only two forces acting on \(P\), find the velocity of \(P\) as it passes through \(B\), giving your answer as a vector. [7]

A particle of mass 0.5 kg is at rest at the point with position vector \((2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})\) m. The particle is then acted upon by two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\). These are the only two forces acting on the particle. Subsequently, the particle passes through the point with position vector \((4\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})\) m with speed 12 m s\(^{-1}\). Given that \(\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j} - \mathbf{k})\) N, find \(\mathbf{F}_2\). [9]

A particle \(P\) projected from a point \(O\) on horizontal ground hits the ground after \(2.4\) seconds. The horizontal component of the initial velocity of \(P\) is \(\frac{5}{3}d \text{ m s}^{-1}\).

1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground. [1]
2. Find the vertical component of the initial velocity of \(P\). [2]

\(P\) just clears a vertical wall which is situated at a horizontal distance \(d\) m from \(O\).

1. Find the height of the wall. [3]

The speed of \(P\) as it passes over the wall is \(16 \text{ m s}^{-1}\).

1. Find the value of \(d\) correct to \(3\) significant figures. [4]

\includegraphics{figure_9} The diagram shows a small block \(B\), of mass \(0.2\) kg, and a particle \(P\), of mass \(0.5\) kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The block can move on the horizontal surface, which is rough. The particle can move on the inclined plane, which is smooth and which makes an angle of \(\theta\) with the horizontal where \(\tan \theta = \frac{3}{4}\). The system is released from rest. In the first \(0.4\) seconds of the motion \(P\) moves \(0.3\) m down the plane and \(B\) does not reach the pulley.

1. Find the tension in the string during the first \(0.4\) seconds of the motion. [4]
2. Calculate the coefficient of friction between \(B\) and the horizontal surface. [5]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) of mass 2 kg is moving on a smooth horizontal surface under the action of a constant horizontal force \((-8\mathbf{i} - 54\mathbf{j})\) N and a variable horizontal force \((4t\mathbf{i} + 6(2t - 1)^2\mathbf{j})\) N.

1. Determine the value of \(t\) when the forces acting on \(P\) are in equilibrium. [2]

It is given that \(P\) is at rest when \(t = 0\).

1. Determine the speed of \(P\) at the instant when \(P\) is moving due north. [6]
2. Determine the distance between the positions of \(P\) when \(t = 0\) and \(t = 3\). [5]

\includegraphics{figure_10} A rectangular block \(B\) is at rest on a horizontal surface. A particle \(P\) of mass 2.5 kg is placed on the upper surface of \(B\). The particle \(P\) is attached to one end of a light inextensible string which passes over a smooth fixed pulley. A particle \(Q\) of mass 3 kg is attached to the other end of the string and hangs freely below the pulley. The part of the string between \(P\) and the pulley is horizontal (see diagram). The particles are released from rest with the string taut. It is given that \(B\) remains in equilibrium while \(P\) moves on the upper surface of \(B\). The tension in the string while \(P\) moves on \(B\) is 16.8 N.

1. Find the acceleration of \(Q\) while \(P\) and \(B\) are in contact. [2]
2. Determine the coefficient of friction between \(P\) and \(B\). [3]
3. Given that the coefficient of friction between \(B\) and the horizontal surface is \(\frac{5}{49}\), determine the least possible value for the mass of \(B\). [3]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle \(P\) is moving on a smooth horizontal surface under the action of a single force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geq 0\), the velocity \(\mathbf{v} \mathrm{m s}^{-1}\) of \(P\), relative to a fixed origin \(O\), is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (2t^2 + t - 13)\mathbf{j}.$$

1. Show that \(P\) is never stationary. [2]
2. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the acceleration of \(P\) at time \(t\). [1]

The mass of \(P\) is 0.5 kg.

1. Determine the magnitude of \(\mathbf{F}\) when \(P\) is moving in the direction of the vector \(-2\mathbf{i} + \mathbf{j}\). Give your answer correct to 3 significant figures. [5]

When \(t = 1\), \(P\) is at the point with position vector \(\frac{1}{6}\mathbf{j}\).

1. Determine the bearing of \(P\) from \(O\) at time \(t = 1.5\). [5]

A particle \(P\) of mass \(m \text{kg}\) is moving on a smooth horizontal surface under the action of two constant horizontal forces \((-4\mathbf{i} + 2\mathbf{j}) \text{N}\) and \((a\mathbf{i} + b\mathbf{j}) \text{N}\). The resultant of these two forces is \(\mathbf{R} \text{N}\). It is given that \(\mathbf{R}\) acts in a direction which is parallel to the vector \(-\mathbf{i} + 3\mathbf{j}\).

1. Show that \(3a + b = 10\). [3]

It is given that \(a = 6\) and that \(P\) moves with an acceleration of magnitude \(5\sqrt{10} \text{ms}^{-2}\).

1. Determine the value of \(m\). [4]

\includegraphics{figure_13} The diagram shows a small block \(B\), of mass \(2 \text{kg}\), and a particle \(P\), of mass \(4 \text{kg}\), which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The particle can move on the inclined plane, which is rough, and which makes an angle of \(60°\) with the horizontal. The block can move on the horizontal surface, which is also rough. The system is released from rest, and in the subsequent motion \(P\) moves down the plane and \(B\) does not reach the pulley. It is given that the coefficient of friction between \(P\) and the inclined plane is twice the coefficient of friction between \(B\) and the horizontal surface.

1. Determine, in terms of \(g\), the tension in the string. [7]

When \(P\) is moving at \(2 \text{ms}^{-1}\) the string breaks. In the \(0.5\) seconds after the string breaks \(P\) moves \(1.9 \text{m}\) down the plane.

1. Determine the deceleration of \(B\) after the string breaks. Give your answer correct to 3 significant figures. [5]

In this question, use \(g = 10\) m s⁻² A box, B, of mass 4 kg lies at rest on a fixed rough horizontal shelf. One end of a light string is connected to B. The string passes over a smooth peg, attached to the end of the shelf. The other end of the string is connected to particle, P, of mass 1 kg, which hangs freely below the shelf as shown in the diagram below. \includegraphics{figure_15} B is initially held at rest with the string taut. B is then released. B and P both move with constant acceleration \(a\) m s⁻² As B moves across the shelf it experiences a total resistance force of 5 N

1. State one type of force that would be included in the total resistance force. [1 mark]
2. Show that \(a = 1\) [4 marks]
3. When B has moved forward exactly 20 cm the string breaks. Find how much further B travels before coming to rest. [4 marks]
4. State one assumption you have made when finding your solutions in parts (b) or (c). [1 mark]