3.03b Newton's first law: equilibrium

3 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the position vector of the particle is \(\mathbf { r }\) metres, where $$\mathbf { r } = \left( 2 \mathrm { e } ^ { \frac { 1 } { 2 } t } - 8 t + 5 \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. 1. Find the speed of the particle when \(t = 3\).
   2. State the direction in which the particle is travelling when \(t = 3\).
3. Find the acceleration of the particle when \(t = 3\).
4. The mass of the particle is 7 kg . Find the magnitude of the resultant force on the particle when \(t = 3\).

5 A particle, of mass 6 kg , is attached to one end of a light inextensible string. The other end of the string is attached to the fixed point \(O\). The particle is set in motion, so that it moves in a horizontal circle at constant speed, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of this circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-4_586_490_541_767} The particle moves in a horizontal circle with an angular speed of 40 revolutions per minute.

1. Show that the angular speed of the particle is \(\frac { 4 \pi } { 3 }\) radians per second.
2. Show that the tension in the string is 67.9 N , correct to three significant figures.
3. Find the radius of the horizontal circle.

4 A particle moves so that at time \(t\) seconds its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) is given by $$\mathbf { v } = \left( 4 t ^ { 3 } - 12 t + 3 \right) \mathbf { i } + 5 \mathbf { j } + 8 t \mathbf { k }$$

1. When \(t = 0\), the position vector of the particle is \(( - 5 \mathbf { i } + 6 \mathbf { k } )\) metres. Find the position vector of the particle at time \(t\).
2. Find the acceleration of the particle at time \(t\).
3. Find the magnitude of the acceleration of the particle at time \(t\). Do not simplify your answer.
4. Hence find the time at which the magnitude of the acceleration is a minimum.
5. The particle is moving under the action of a single variable force \(\mathbf { F }\) newtons. The mass of the particle is 7 kg . Find the minimum magnitude of \(\mathbf { F }\).

1 A particle moves in a straight line and at time \(t\) seconds has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 6 t ^ { 2 } + 4 t - 7 , \quad t \geqslant 0$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The mass of the particle is 3 kg . Find the resultant force on the particle when \(t = 4\).
3. When \(t = 0\), the displacement of the particle from the origin is 5 metres. Find an expression for the displacement of the particle from the origin at time \(t\).

4 A van, of mass 1500 kg , has a maximum speed of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal road. When the van travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(40 v\) newtons.

1. Show that the maximum power of the van is 100000 watts.
2. The van is travelling along a straight horizontal road. Find the maximum possible acceleration of the van when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The van starts to climb a hill which is inclined at \(6 ^ { \circ }\) to the horizontal. Find the maximum possible constant speed of the van as it travels in a straight line up the hill.
   (6 marks)

15 A cyclist is towing a trailer behind her bicycle. She is riding along a straight, horizontal path at a constant speed. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-22_371_723_447_657} A tension of \(T\) newtons acts on the connecting rod between the bicycle and the trailer.
The cyclist is causing a constant driving force of 40 N to be applied whilst pedalling forwards on her bicycle. The constant resistance force acting on the trailer is 12 N
15

1. State the value of \(T\) giving a clear reason for your answer.
   15
2. State one assumption you have made in reaching your answer to part (a).

9. \begin{figure}[h] \end{figure} Two small balls, \(P\) and \(Q\), have masses \(2 m\) and \(k m\) respectively, where \(k < 2\).
The balls are attached to the ends of a string that passes over a fixed pulley.
The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The system is released from rest and, in the subsequent motion, \(P\) moves downwards with an acceleration of magnitude \(\frac { 5 g } { 7 }\) The balls are modelled as particles moving freely.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Using the model,

1. find, in terms of \(m\) and \(g\), the tension in the string,
2. explain why the acceleration of \(Q\) also has magnitude \(\frac { 5 g } { 7 }\)
3. find the value of \(k\).
4. Identify one limitation of the model that will affect the accuracy of your answer to part (c).

9. \begin{figure}[h] \end{figure} A small ball \(A\) of mass 2.5 kg is held at rest on a rough horizontal table.
The ball is attached to one end of a string.
The string passes over a pulley \(P\) which is fixed at the edge of the table. The other end of the string is attached to a small ball \(B\) of mass 1.5 kg hanging freely, vertically below \(P\) and with \(B\) at a height of 1 m above the horizontal floor. The system is release from rest, with the string taut, as shown in Figure 2.
The resistance to the motion of \(A\) from the rough table is modelled as having constant magnitude 12.7 N . Ball \(B\) reaches the floor before ball \(A\) reaches the pulley. The balls are modelled as particles, the string is modelled as being light and inextensible, the pulley is modelled as being small and smooth and the acceleration due to gravity, \(g\), is modelled as being \(9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. 1. Write down an equation of motion for \(A\).
   2. Write down an equation of motion for \(B\).
2. Hence find the acceleration of \(B\).
3. Using the model, find the time it takes, from release, for \(B\) to reach the floor.
4. Suggest two improvements that could be made in the model.

10 A small body of mass \(m\) is thrown vertically upwards with initial velocity \(u\). Resistance to motion is \(k v ^ { 2 }\) per unit mass, where the velocity is \(v\) and \(k\) is a positive constant. Find, in terms of \(u , g\) and \(k\),

1. the time taken to reach the greatest height,
2. the greatest height to which the body will rise.

10 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-6_490_661_267_703} Particles \(A\) and \(B\) of masses \(2 m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45 ^ { \circ }\) and \(B\) is above the plane. The vertical plane defined by \(A P B\) contains a line of greatest slope of the plane, and \(P A\) is inclined at angle \(2 \alpha\) to the horizontal (see diagram).

1. Show that the normal reaction \(R\) between \(A\) and the plane is \(m g ( 2 \cos \alpha - \sin \alpha )\).
2. Show that \(R \geqslant \frac { 1 } { 2 } m g \sqrt { 2 }\). The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.
3. Show that \(0.5 < \tan \alpha \leqslant 1\).
4. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies.

\includegraphics{figure_2} Coplanar forces of magnitudes 20 N, \(P\) N, \(3P\) N and \(4P\) N act at a point in the directions shown in the diagram. The system is in equilibrium. Find \(P\) and \(\theta\). [6]

\includegraphics{figure_4} Three coplanar forces of magnitudes 20 N, 100 N and \(F\) N act at a point. The directions of these forces are shown in the diagram. Given that the three forces are in equilibrium, find \(F\) and \(\alpha\). [6]

\includegraphics{figure_1} A block \(B\) of mass 5 kg is attached to one end of a light inextensible string. A particle \(P\) of mass 4 kg is attached to other end of the string. The string passes over a smooth pulley. The system is in equilibrium with the string taut and its straight parts vertical. \(B\) is at rest on the ground (see diagram). State the tension in the string and find the force exerted on \(B\) by the ground. [3]

\includegraphics{figure_3} A particle is moving under the action of three forces as shown in the diagram. The particle is in equilibrium. Find the magnitudes of forces \(P\) and \(Q\). [6]

\includegraphics{figure_2} The diagram shows three coplanar forces acting at the point \(O\). The magnitudes of the forces are \(6 \text{ N}\), \(8 \text{ N}\) and \(10 \text{ N}\). The angle between the \(6 \text{ N}\) force and the \(8 \text{ N}\) force is \(90°\). The forces are in equilibrium. Find the other angles between the forces. [4]

\includegraphics{figure_5} A small ring \(P\) is threaded on a fixed smooth horizontal rod \(AB\). Three horizontal forces of magnitudes 4.5 N, 7.5 N and \(F\) N act on \(P\) (see diagram).

1. Given that these three forces are in equilibrium, find the values of \(F\) and \(\theta\). [6]

1. It is given instead that the values of \(F\) and \(\theta\) are 9.5 and 30 respectively, and the acceleration of the ring is 1.5 m s\(^{-2}\). Find the mass of the ring. [2]

\includegraphics{figure_3} A particle \(P\) of mass 0.3 kg is held in equilibrium above a horizontal plane by a force of magnitude 5 N, acting vertically upwards. The particle is attached to two strings \(PA\) and \(PB\) of lengths 0.9 m and 1.2 m respectively. The points \(A\) and \(B\) lie on the plane and angle \(APB = 90°\) (see diagram). Find the tension in each of the strings. [5]

\includegraphics{figure_5} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(AB\) of length 3 m. Three horizontal forces of magnitudes \(F\) N, 10 N and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R\) N in the direction shown in the diagram.

1. Find the values of \(F\) and \(R\). [5]
2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of 11.7 m s\(^{-1}\). Find the mass of the bead. [3]

\includegraphics{figure_1} A particle of weight 24 N is held in equilibrium by two light inextensible strings. One string is horizontal. The other string is inclined at an angle of 30° to the horizontal, as shown in Figure 1. The tension in the horizontal string is \(Q\) newtons and the tension in the other string is \(P\) newtons. Find

1. the value of \(P\), [3]
2. the value of \(Q\). [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_1} A smooth bead \(B\) is threaded on a light inextensible string. The ends of the string are attached to two fixed points \(A\) and \(C\) on the same horizontal level. The bead is held in equilibrium by a horizontal force of magnitude 6 N acting parallel to \(AC\). The bead \(B\) is vertically below \(C\) and \(\angle BAC = \alpha\), as shown in Figure 1. Given that \(\tan \alpha = \frac{3}{4}\), find

1. the tension in the string, [3]
2. the weight of the bead. [4]

\includegraphics{figure_2} A box of mass 2 kg is pulled up a rough plane face by means of a light rope. The plane is inclined at an angle of \(20°\) to the horizontal, as shown in Figure 2. The rope is parallel to a line of greatest slope of the plane. The tension in the rope is 18 N. The coefficient of friction between the box and the plane is 0.6. By modelling the box as a particle, find

1. the normal reaction of the plane on the box, [3]
2. the acceleration of the box. [5]

\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 small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{4}{3}\), and released from rest. The coefficient of friction between the brick and the plane is \(\frac{1}{3}\). Find the acceleration of the brick. [9]

\includegraphics{figure_1} A small box of mass 15 kg rests on a rough horizontal plane. The coefficient of friction between the box and the plane is 0.2. A force of magnitude \(P\) newtons is applied to the box at 50° to the horizontal, as shown in Figure 1. The box is on the point of sliding along the plane. Find the value of \(P\), giving your answer to 2 significant figures. [9]