3.03d Newton's second law: 2D vectors

2 A particle moves in a straight line. At time \(t\) seconds, it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 6 t ^ { 2 } - 2 \mathrm { e } ^ { - 4 t } + 8$$ and \(t \geqslant 0\).

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. Find the acceleration of the particle when \(t = 0.5\).
2. The particle has mass 4 kg . Find the magnitude of the force acting on the particle when \(t = 0.5\).
3. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).

1 A particle, of mass 4 kg , moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal plane, perpendicular to each other. At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = 4 \cos 2 t \mathbf { i } + 3 \sin t \mathbf { j }$$

1. 1. Find an expression for the force, \(\mathbf { F }\), acting on the particle at time \(t\) seconds.
   2. Find the magnitude of \(\mathbf { F }\) when \(t = \pi\).
2. When \(t = 0\), the particle is at the point with position vector \(( 2 \mathbf { i } - 14 \mathbf { j } )\) metres. Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\) seconds.
   [0pt] [5 marks]
   \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-02_1346_1717_1361_150}

4

1. A parachutist and her equipment have a combined mass of 80 kg . During a descent where the parachutist loses 1600 m in height, her speed reduces from \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and she does \(1.3 \times 10 ^ { 6 } \mathrm {~J}\) of work against resistances. Use an energy method to calculate the value of \(V\).
2. A vehicle of mass 800 kg is climbing a hill inclined at \(\theta\) to the horizontal, where \(\sin \theta = 0.1\). At one time the vehicle has a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is accelerating up the hill at \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) against a resistance of 1150 N .
   1. Show that the driving force on the vehicle is 2134 N and calculate its power at this time. The vehicle is pulling a sledge, of mass 300 kg , which is sliding up the hill. The sledge is attached to the vehicle by a light, rigid coupling parallel to the slope. The force in the coupling is 900 N .
   2. Assuming that the only resistance to the motion of the sledge is due to friction, calculate the coefficient of friction between the sledge and the ground.

1. A particle \(P\) of mass 2 kg is subjected to a force \(\mathbf { F }\) such that its displacement, \(\mathbf { r }\) metres, from a fixed origin, \(O\), at time \(t\) seconds is given by

$$\mathbf { r } = \left( 3 t ^ { 2 } - 4 \right) \mathbf { i } + \left( 3 - 4 t ^ { 2 } \right) \mathbf { j }$$

1. Show that the acceleration of \(P\) is constant.
2. Find the magnitude of \(\mathbf { F }\).
   

4 A smooth cylinder of radius \(a \mathrm {~m}\) is fixed with its axis horizontal and \(O\) is the centre of a cross-section. Particle \(P\), of mass 0.4 kg , and particle \(Q\), of mass 0.6 kg , are connected by a light inextensible string of length \(\pi a \mathrm {~m}\). The string is held at rest with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the crosssection through \(O\) (see Fig. 1). The string is released and \(Q\) begins to descend. When \(O P\) has rotated through \(\theta\) radians, with \(P\) remaining in contact with the cylinder, the speed of each particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 3.92 a ( 3 \theta - 2 \sin \theta )\) and find an expression in terms of \(\theta\) for the normal force of the cylinder on \(P\) at this time.
2. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(\sin \alpha = k \alpha\) where \(k\) is a constant to be found.

2

1. A fixed solid sphere with a smooth surface has centre O and radius 0.8 m . A particle P is given a horizontal velocity of \(1.2 \mathrm {~ms} ^ { - 1 }\) at the highest point on the sphere, and it moves on the surface of the sphere in part of a vertical circle of radius 0.8 m .
   1. Find the radial and tangential components of the acceleration of P at the instant when OP makes an angle \(\frac { 1 } { 6 } \pi\) radians with the upward vertical. (You may assume that P is still in contact with the sphere.)
   2. Find the speed of P at the instant when it leaves the surface of the sphere.
2. Two fixed points R and S are 2.5 m apart with S vertically below R . A particle Q of mass 0.9 kg is connected to R and to S by two light inextensible strings; Q is moving in a horizontal circle at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with both strings taut. The radius of the circle is 2.4 m and the centre C of the circle is 0.7 m vertically below S, as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3f674569-7e99-4ba8-84f1-a1eb438e30ed-2_547_720_1946_644} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Find the tension in the string RQ and the tension in the string \(S Q\).

2

1. A particle P of mass 0.6 kg is connected to a fixed point by a light inextensible string of length 2.8 m . The particle P moves in a horizontal circle as a conical pendulum, with the string making a constant angle of \(55 ^ { \circ }\) with the vertical.
   1. Find the tension in the string.
   2. Find the speed of P .
2. A turntable has a rough horizontal surface, and it can rotate about a vertical axis through its centre O . While the turntable is stationary, a small object Q of mass 0.5 kg is placed on the turntable at a distance of 1.4 m from O . The turntable then begins to rotate, with a constant angular acceleration of \(1.12 \mathrm { rad } \mathrm { s } ^ { - 2 }\). Let \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) be the angular speed of the turntable. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-3_517_522_870_769} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Given that Q does not slip, find the components \(F _ { 1 }\) and \(F _ { 2 }\) of the frictional force acting on Q perpendicular and parallel to QO (see Fig. 2). Give your answers in terms of \(\omega\) where appropriate. The coefficient of friction between Q and the turntable is 0.65 .
   2. Find the value of \(\omega\) when Q is about to slip.
   3. Find the angle which the frictional force makes with QO when Q is about to slip.

1 A fixed solid sphere has centre O and radius 2.6 m . A particle P of mass 0.65 kg moves on the smooth surface of the sphere. The particle P is set in motion with horizontal velocity \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the highest point of the sphere, and moves in part of a vertical circle. When OP makes an angle \(\theta\) with the upward vertical, and P is still in contact with the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 52.92 - 50.96 \cos \theta\).
2. Find, in terms of \(\theta\), the normal reaction acting on P .
3. Find the speed of P at the instant when it leaves the surface of the sphere. The particle P is now attached to one end of a light inextensible string, and the other end of the string is fixed to a point A , vertically above O , such that AP is tangential to the sphere, as shown in Fig. 1. P moves with constant speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle with radius 2.4 m on the surface of the sphere. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3ec81c4e-e0fa-43d9-9c79-ef9df746be8f-2_1100_634_1089_753} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
4. Find the tension in the string and the normal reaction acting on P .

2

1. A particle P of mass 0.2 kg is connected to a fixed point O by a light inextensible string of length 3.2 m , and is moving in a vertical circle with centre O and radius 3.2 m . Air resistance may be neglected. When P is at the highest point of the circle, the tension in the string is 0.6 N .
   1. Find the speed of P when it is at the highest point.
   2. For an instant when OP makes an angle of \(60 ^ { \circ }\) with the downward vertical, find
      (A) the radial and tangential components of the acceleration of P ,
      (B) the tension in the string.
2. A solid cone is fixed with its axis of symmetry vertical and its vertex V uppermost. The semivertical angle of the cone is \(36 ^ { \circ }\), and its surface is smooth. A particle Q of mass 0.2 kg is connected to V by a light inextensible string, and Q moves in a horizontal circle at constant speed, in contact with the surface of the cone, as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-3_455_609_950_808} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The particle Q makes one complete revolution in 1.8 s , and the normal reaction of the cone on Q has magnitude 0.75 N .
   1. Find the tension in the string.
   2. Find the length of the string.

2. A particle \(P\) of mass 0.25 kg is moving on a horizontal plane. At time \(t\) seconds the velocity, \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), of \(P\) relative to a fixed origin \(O\) is given by $$\mathbf { v } = \ln ( t + 1 ) \mathbf { i } - \mathrm { e } ^ { - 2 t } \mathbf { j } , t \leq 0 ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane.

1. Find the acceleration of \(P\) in terms of \(t\).
2. Find, correct to 3 significant figures, the magnitude of the resultant force acting on \(P\) when \(t = 1\).
   (4 marks)

1. At time \(t = 0\), a particle \(P\) of mass 3 kg is at rest at the point \(A\) with position vector \(( \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }\). 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 \(( 8 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k } ) \mathrm { m }\).

Given that \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( 8 \mathbf { i } - 4 \mathbf { j } + 7 \mathbf { k } ) \mathrm { N }\) 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.

6. A uniform circular disc, of radius \(r\) and mass \(m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which is perpendicular to the plane of the disc and passes through a point which is \(\frac { 1 } { 4 } r\) from the centre of the disc. The disc is held at rest with its centre vertically above the axis. The disc is then slightly disturbed from its rest position. You may assume without proof that the moment of inertia of the disc about \(L\) is \(\frac { 9 m r ^ { 2 } } { 16 }\).

1. Show that the angular speed of the disc when it has turned through \(\frac { \pi } { 2 }\) is \(\sqrt { } \left( \frac { 8 g } { 9 r } \right)\).
2. Find the magnitude of the force exerted on the disc by the axis when the disc has turned through \(\frac { \pi } { 2 }\).

7. A uniform circular disc, of radius \(r\) and mass \(m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis. This axis is perpendicular to the plane of the disc and passes through a point \(A\) on the circumference of the disc. The disc is held with \(A B\) horizontal, where \(A B\) is a diameter of the disc, and released from rest.

1. Find the magnitude of
   1. the horizontal component,
   2. the vertical component
      of the force exerted on the disc by the axis immediately after the disc is released. When \(A B\) is vertical the disc is instantaneously brought to rest by a horizontal impulse which acts in the plane of the disc and is applied to the disc at \(B\).
2. Find the magnitude of the impulse.

1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are acting on an object of mass 3 kg such that

$$\begin{aligned} & \mathbf { F } _ { 1 } = ( \mathbf { i } + 8 c \mathbf { j } + 11 c \mathbf { k } ) \mathrm { N } , \\ & \mathbf { F } _ { 2 } = ( - 14 \mathbf { i } - c \mathbf { j } - 12 \mathbf { k } ) \mathrm { N } , \\ & \mathbf { F } _ { 3 } = ( ( 15 c + 1 ) \mathbf { i } + 2 c \mathbf { j } - 5 c \mathbf { k } ) \mathrm { N } , \end{aligned}$$ where \(c\) is a constant. The acceleration of the object is parallel to the vector \(( \mathbf { i } + \mathbf { j } )\).

1. Find the value of the constant \(c\) and hence show that the acceleration of the object is \(( 6 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\).
   
2. When \(t = 0\) seconds, the object has position vector \(\mathbf { r } _ { 0 } \mathrm {~m}\) and is moving with velocity \(( - 17 \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When \(t = 4\) seconds, the object has position vector \(( - 13 \mathbf { i } + 84 \mathbf { j } ) \mathrm { m }\). Find the vector \(\mathbf { r } _ { 0 }\).
   

5. A particle of mass 2 kg is moving under the action of a force \(\mathbf { F N }\) which, at time \(t\) seconds, is given by $$\mathbf { F } = 4 t \mathbf { i } - \sqrt { t } \mathbf { j } + 6 \mathbf { k }$$ When \(t = 1\), the velocity of the particle is \(\left( 3 \mathbf { i } - \frac { 1 } { 3 } \mathbf { j } - \mathbf { k } \right) \mathrm { ms } ^ { - 1 }\).

1. Find an expression for the velocity vector of the particle at time \(t \mathrm {~s}\).
2. Determine the values of \(t\) when the particle is moving in a direction perpendicular to the vector \(( - \mathbf { i } + 3 \mathbf { k } )\).

1. A racing car of mass 750 kg is moving along a straight horizontal road at a constant speed of \(U \mathbf { k m ~ h } ^ { - \mathbf { 1 } }\). The engine of the racing car is working at a constant rate of 60 kW .

The resistance to the motion of the racing car is modelled as a force of magnitude \(37.5 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the racing car. Using the model,

1. find the value of \(U\) Later on, the racing car is accelerating up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 5 } { 49 }\). The engine of the racing car is working at a constant rate of 60 kW . The total resistance to the motion of the racing car from non-gravitational forces is modelled as a force of magnitude \(37.5 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the racing car. At the instant when the acceleration of the racing car is \(2 \mathrm {~ms} ^ { - 2 }\), the speed of the racing car is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the model,
2. find the value of \(V\)

1. A lorry has mass 5000 kg .

In all circumstances, when the speed of the lorry is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to motion of the lorry from non-gravitational forces is modelled as having magnitude \(490 v\) newtons. The lorry moves along a straight horizontal road at \(12 \mathrm {~ms} ^ { - 1 }\), with its engine working at a constant rate of 84 kW . Using the model,

1. find the acceleration of the lorry. Another straight road is inclined to the horizontal at an angle \(\alpha\) where \(\sin \alpha = \frac { 1 } { 14 }\) With its engine again working at a constant rate of 84 kW , the lorry can maintain a constant speed of \(V \mathrm {~ms} ^ { - 1 }\) up the road. Using the model,
2. find the value of \(V\).

1. \hspace{0pt} [In this question use \(\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) ]

A jogger of mass 60 kg runs along a straight horizontal road at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The total resistance to the motion of the jogger is modelled as a constant force of magnitude 30 N .

1. Find the rate at which the jogger is working. The jogger now comes to a hill which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\). Because of the hill, the jogger reduces her speed to \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and maintains this constant speed as she runs up the hill. The total resistance to the motion of the jogger from non-gravitational forces continues to be modelled as a constant force of magnitude 30 N .
2. Find the rate at which she has to work in order to run up the hill at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2. \begin{figure}[h] \end{figure} A thin hollow hemisphere, with centre \(O\) and radius \(a\), is fixed with its axis vertical, as shown in Figure 2. A small ball \(B\) of mass \(m\) moves in a horizontal circle on the inner surface of the hemisphere. The circle has centre \(C\) and radius \(r\). The point \(C\) is vertically below \(O\) such that \(O C = h\). The ball moves with constant angular speed \(\omega\) The inner surface of the hemisphere is modelled as being smooth and \(B\) is modelled as a particle. Air resistance is modelled as being negligible.

1. Show that \(\omega ^ { 2 } = \frac { g } { h }\) Given that the magnitude of the normal reaction between \(B\) and the surface of the hemisphere is \(3 m g\)
2. find \(\omega\) in terms of \(g\) and \(a\).
3. State how, apart from ignoring air resistance, you have used the fact that \(B\) is modelled as a particle.

A light inextensible string has length 7a. One end of the string is attached to a fixed point \(A\) and the other end of the string is attached to a fixed point \(B\), with \(A\) vertically above \(B\) and \(A B = 5 a\). A particle of mass \(m\) is attached to a point \(P\) on the string where \(A P = 4 a\). The particle moves in a horizontal circle with constant angular speed \(\omega\), with both \(A P\) and \(B P\) taut.

1. Show that
   1. the tension in \(A P\) is \(\frac { 4 m } { 25 } \left( 9 a \omega ^ { 2 } + 5 g \right)\)
   2. the tension in \(B P\) is \(\frac { 3 m } { 25 } \left( 16 a \omega ^ { 2 } - 5 g \right)\). The string will break if the tension in it reaches a magnitude of \(4 m g\).
      The time for the particle to make one revolution is \(S\).
2. Show that $$3 \pi \sqrt { \frac { a } { 5 g } } < S < 8 \pi \sqrt { \frac { a } { 5 g } }$$
3. State how in your calculations you have used the assumption that the string is light.

2. \begin{figure}[h] \end{figure} A van of mass 600 kg is moving up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\). The van is towing a trailer of mass 150 kg . The van is attached to the trailer by a towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 1. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 200 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 100 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 12 kW .
Find the tension in the towbar at the instant when the speed of the van is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car is a constant force of magnitude 400 N.

The engine of the car is working at a constant rate of 16 kW .

1. Find the value of \(U\). The car now pulls a trailer of mass 600 kg in a straight line along the road using a tow rope which is parallel to the direction of motion. The resistance to the motion of the car is again a constant force of magnitude 400 N . The resistance to the motion of the trailer is a constant force of magnitude 300 N . The engine of the car is working at a constant rate of 16 kW .
   The tow rope is modelled as being light and inextensible.
   Using the model,
2. find the tension in the tow rope at the instant when the speed of the car is \(\frac { 20 } { 3 } \mathrm {~ms} ^ { - 1 }\)

1. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)

• The resistance to the motion of the car is modelled as a constant force of magnitude 900 N

The engine of the car is working at a constant rate of \(P \mathrm {~kW}\).
Using the model,

1. find the value of \(P\). The car now travels in a straight line up a road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 2 } { 49 }\)
   At the instant when the engine of the car is working at a constant rate of 30 kW and the car is moving up the road at \(10 \mathrm {~ms} ^ { - 1 }\), the acceleration of the car is \(a \mathrm {~ms} ^ { - 2 }\) Using the refined model,
2. find the value of \(a\). Later on, when the engine of the car is again working at a constant rate of 30 kW , the car is moving up the road at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the refined model,
3. find the value of \(U\).

1. A car of mass 600 kg is moving along a straight horizontal road.

At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car is modelled as a force of magnitude \(( 200 + 2 v ) \mathrm { N }\). The engine of the car is working at a constant rate of 12 kW .

1. Find the acceleration of the car at the instant when \(v = 20\) Later on the car is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\) At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude ( \(200 + 2 v ) \mathrm { N }\). The engine is again working at a constant rate of 12 kW .
   At the instant when the car has speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car is decelerating at \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(w\).

1. \begin{figure}[h] \end{figure} A hemispherical shell of radius \(a\) is fixed with its rim uppermost and horizontal. A small bead, \(B\), is moving with constant angular speed, \(\omega\), in a horizontal circle on the smooth inner surface of the shell. The centre of the path of \(B\) is at a distance \(\frac { 1 } { 4 } a\) vertically below the level of the rim of the hemisphere, as shown in Figure 1. Find the magnitude of \(\omega\), giving your answer in terms of \(a\) and \(g\).