3.03d Newton's second law: 2D vectors

1 A car of mass 800 kg is moving on a straight horizontal road with its engine working at a rate of 22.5 kW . Find the resistance to the car's motion at an instant when the car's speed is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

5
A toy aircraft of mass 0.5 kg is attached to one end of a light inextensible string of length 9 m . The other end of the string is attached to a fixed point \(O\). The aircraft moves with constant speed in a horizontal circle. The string is taut, and makes an angle of \(60 ^ { \circ }\) with the upward vertical at \(O\) (see diagram). In a simplified model of the motion, the aircraft is treated as a particle and the force of the air on the aircraft is taken to act vertically upwards with magnitude 8 N . Find

1. the tension in the string,
2. the speed of the aircraft.

7 One end of a light inextensible string of length 0.15 m is attached to a fixed point which is above a smooth horizontal surface. A particle of mass 0.5 kg is attached to the other end of the string. The particle moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, with the string taut and making an angle of \(\theta ^ { \circ }\) with the downward vertical.

1. Given that \(\theta = 60\) and that the particle is not in contact with the surface, find \(v\).
2. Given instead that \(\theta = 45\) and \(v = 0.9\), and that the particle is in contact with the surface, find
   1. the tension in the string,
   2. the force exerted by the surface on the particle.

2 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-2_561_570_1274_790} A particle of mass 0.15 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The magnitude of the acceleration of the particle is \(7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The string makes an angle of \(\theta ^ { \circ }\) with the downward vertical, as shown in the diagram. Find

1. the value of \(\theta\) to the nearest whole number,
2. the tension in the string,
3. the speed of the particle.

3 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A hollow container consists of a smooth circular cylinder of radius 0.5 m , and a smooth hollow cone of semi-vertical angle \(65 ^ { \circ }\) and radius 0.5 m . The container is fixed with its axis vertical and with the cone below the cylinder. A steel ball of weight 1 N moves with constant speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle inside the container. The ball is in contact with both the cylinder and the cone (see Fig. 1). Fig. 2 shows the forces acting on the ball, i.e. its weight and the forces of magnitudes \(R \mathrm {~N}\) and \(S \mathrm {~N}\) exerted by the container at the points of contact. Given that the radius of the ball is negligible compared with the radius of the cylinder, find \(R\) and \(S\).

3 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-3_637_572_264_788} One end of a light inextensible string is attached to a point \(C\). The other end is attached to a point \(D\), which is 1.1 m vertically below \(C\). A small smooth ring \(R\), of mass 0.2 kg , is threaded on the string and moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, with centre at \(O\) and radius 1.2 m , where \(O\) is 0.5 m vertically below \(D\) (see diagram).

1. Show that the tension in the string is 1.69 N , correct to 3 significant figures.
2. Find the value of \(v\).

4
A particle of mass 0.12 kg is moving on the smooth inside surface of a fixed hollow sphere of radius 0.5 m . The particle moves in a horizontal circle whose centre is 0.3 m below the centre of the sphere (see diagram).

1. Show that the force exerted by the sphere on the particle has magnitude 2 N .
2. Find the speed of the particle.
3. Find the time taken for the particle to complete one revolution.

3 \includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-3_456_511_260_817} A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T \mathrm {~N}\). The acceleration of the particle has magnitude \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(\tan \theta = 0.75\) and find the value of \(T\).
2. Find the speed of the particle.

3 \includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-3_385_1154_253_497} A particle \(P\) of mass 0.5 kg is attached to the vertex \(V\) of a fixed solid cone by a light inextensible string. \(P\) lies on the smooth curved surface of the cone and moves in a horizontal circle of radius 0.1 m with centre on the axis of the cone. The cone has semi-vertical angle \(60 ^ { \circ }\) (see diagram).

1. Calculate the speed of \(P\), given that the tension in the string and the contact force between the cone and \(P\) have the same magnitude.
2. Calculate the greatest angular speed at which \(P\) can move on the surface of the cone.

1 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-2_381_1079_255_534} A particle \(P\) of mass 0.4 kg is attached to a fixed point \(A\) by a light inextensible string. The string is inclined at \(60 ^ { \circ }\) to the vertical. \(P\) moves with constant speed in a horizontal circle of radius 0.2 m . The centre of the circle is vertically below \(A\) (see diagram).

1. Show that the tension in the string is 8 N .
2. Calculate the speed of the particle.

3 A small block \(B\) of mass 0.2 kg is placed at a fixed point \(O\) on a smooth horizontal surface. A horizontal force of magnitude 0.42 N is applied to \(B\). At time \(t \mathrm {~s}\) after the force is first applied, the velocity of \(B\) away from \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(v\) when \(t = 1\). For \(t > 1\) an additional force, of magnitude \(0.32 t \mathrm {~N}\) and directed towards \(O\), is applied to \(B\). The force of magnitude 0.42 N continues to act as before.
2. Find the value of \(v\) when \(t = 2\). For \(t > 2\) a third force, of magnitude \(0.06 t ^ { 2 } \mathrm {~N}\) and directed away from \(O\), is applied to \(B\). The other two forces continue to act as before.
3. Show that the velocity of \(B\) is the same when \(t = 2\) and when \(t = 3\).

6 A cyclist and her bicycle have a total mass of 60 kg . The cyclist rides in a horizontal straight line, and exerts a constant force in the direction of motion of 150 N . The motion is opposed by a resistance of magnitude \(12 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the cyclist's speed at time \(t \mathrm {~s}\) after passing through a fixed point \(A\).

1. Show that \(5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = 12.5 - v\).
2. Given that the cyclist passes through \(A\) with speed \(11.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), solve this differential equation to show that \(v = 12.5 - \mathrm { e } ^ { - 0.2 t }\).
3. Express the displacement of the cyclist from \(A\) in terms of \(t\).

4 A particle of mass 0.4 kg is released from rest and falls vertically. A resisting force of magnitude \(0.08 v \mathrm {~N}\) acts upwards on the particle during its descent, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the particle at time \(t \mathrm {~s}\) after its release.

1. Show that the acceleration of the particle is \(( 10 - 0.2 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find the velocity of the particle when \(t = 15\).

3 A particle \(P\) of mass 0.5 kg moves along the \(x\)-axis on a horizontal surface. When the displacement of \(P\) from the origin \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Two horizontal forces act on \(P\); one force has magnitude \(\left( 1 + 0.3 x ^ { 2 } \right) \mathrm { N }\) and acts in the positive \(x\)-direction, and the other force has magnitude \(8 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the negative \(x\)-direction.

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 2 + 0.6 x ^ { 2 } - 16 \mathrm { e } ^ { - x }\).
2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~ms} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 3\).
3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-3_259_745_278_740} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A small sphere \(A\) of mass 0.15 kg is moving inside a fixed smooth hollow cylinder whose axis is vertical. \(A\) moves with constant speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle of radius 0.35 m , and is continuously in contact with both the plane base and the curved surface of the cylinder. Fig. 1 shows a vertical cross-section of the cylinder through its axis. Find the magnitude of the force exerted on \(A\) by

4 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string and is held with the string taut at the point \(A\). At \(A\) the string makes an angle \(\theta\) with the upward vertical through \(O\). The particle is projected perpendicular to the string in a downward direction from \(A\) with a speed \(u\). It moves along a circular path in the vertical plane. When the string makes an angle \(\alpha\) with the downward vertical through \(O\), the speed of the particle is \(2 u\) and the magnitude of the tension in the string is 10 times its magnitude at \(A\). It is given that \(\mathrm { u } = \sqrt { \frac { 2 } { 3 } \mathrm { ga } }\).

1. Find, in terms of \(m\) and \(g\), the magnitude of the tension in the string at \(A\).
2. Find the value of \(\cos \alpha\).

5 A small ball \(B\) of mass 0.2 kg is attached to fixed points \(P\) and \(Q\) by two light inextensible strings of equal length. \(P\) is vertically above \(Q\), the strings are taut and each is inclined at \(60 ^ { \circ }\) to the vertical. \(B\) moves with constant speed in a horizontal circle of radius 0.6 m .

1. Given that the tension in the string \(P B\) is 7 N , calculate
   1. the tension in string \(Q B\),
   2. the speed of \(B\).
   3. Given instead that \(B\) is moving with angular speed \(7 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the tension in the string \(Q B\).

1 A particle \(P\) of mass 0.3 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point \(O\) of a smooth horizontal plane. \(P\) moves on the plane at constant speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a circle with centre \(O\). Calculate the tension in the string.

3 A particle \(P\) of mass 0.8 kg moves along the \(x\)-axis on a horizontal surface. When the displacement of \(P\) from the origin \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Two horizontal forces act on \(P\). One force has magnitude \(4 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the positive \(x\)-direction. The other force has magnitude \(2.4 x ^ { 2 } \mathrm {~N}\) and acts in the negative \(x\)-direction.

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 \mathrm { e } ^ { - x } - 3 x ^ { 2 }\).
2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 2\).

5 \includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-3_540_537_255_804} A particle \(P\) of mass 0.2 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.4 m . A second light inextensible string of length 0.3 m connects \(P\) to a fixed point \(B\) which is vertically below \(A\). The particle \(P\) moves in a horizontal circle, which has its centre on the line \(A B\), with the angle \(A P B = 90 ^ { \circ }\) (see diagram).

1. Given that the tensions in the two strings are equal, calculate the speed of \(P\).
2. It is given instead that \(P\) moves with its least possible angular speed for motion in this circle. Find this angular speed.

3 A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 12 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is projected vertically downwards with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the point 0.5 m vertically below \(O\). For an instant when the acceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards, find the extension of the string and the speed of \(P\).

5 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(O A\) is a line of greatest slope of the plane with \(A\) below the level of \(O\) and \(O A = 0.8 \mathrm {~m}\). The particle \(P\) is released from rest at \(A\).

1. Find the initial acceleration of \(P\).
2. Find the greatest speed of \(P\). \(6 \quad A\) and \(B\) are two fixed points on a vertical axis with \(A 0.6 \mathrm {~m}\) above \(B\). A particle \(P\) of mass 0.3 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by a light elastic string with modulus of elasticity 46 N . The particle \(P\) moves with constant angular speed \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with centre at the mid-point of \(A B\).
3. Find the speed of \(P\).
4. Calculate the tension in the string \(B P\) and hence find the natural length of this string. \includegraphics[max width=\textwidth, alt={}, center]{42de91da-d65e-40e7-8de5-f40eda927850-10_540_574_260_781} \(A B C\) is the cross-section through the centre of mass of a uniform prism which rests with \(A B\) on a rough horizontal surface. \(A B = 0.4 \mathrm {~m}\) and \(C\) is 0.9 m above the surface (see diagram). The prism is on the point of toppling about its edge through \(B\).
5. Show that angle \(B A C = 48.4 ^ { \circ }\), correct to 3 significant figures.
   A force of magnitude 18 N acting in the plane of the cross-section and perpendicular to \(A C\) is now applied to the prism at \(C\). The prism is on the point of rotating about its edge through \(A\).
6. Calculate the weight of the prism.
7. Given also that the prism is on the point of slipping, calculate the coefficient of friction between the prism and the surface.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 \includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-05_392_621_255_762} A particle \(P\) of mass 0.4 kg moves with constant speed in a horizontal circle on the smooth inner surface of a fixed hollow hemisphere with centre \(O\) and radius 0.5 m (see diagram).

1. Given that the speed of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular speed is \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the angle between \(O P\) and the vertical.
2. Given instead that the magnitude of the force exerted on \(P\) by the hemisphere is 6 N , calculate
   1. the angle between \(O P\) and the vertical,
   2. the angular speed of \(P\).

7 A particle \(P\) of mass \(M \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 12.5 N . The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and falls vertically until it comes to instantaneous rest at the point \(B\). The greatest speed of \(P\) during its descent is \(4.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(e \mathrm {~m}\).

1. Show that \(e = 0.64 M\).
2. Find a second equation in \(e\) and \(M\), and hence find \(M\).
3. Calculate the distance \(A B\).

1 A particle \(P\) of mass \(m\) is placed on a fixed smooth plane which is inclined at an angle \(\theta\) to the horizontal. A light spring, of natural length \(a\) and modulus of elasticity \(3 m g\), has one end attached to \(P\) and the other end attached to a fixed point \(O\) at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length. Find, in terms of \(a\) and \(\theta\), an expression for the greatest extension of the spring in the subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{1c53c407-25ea-43fc-a571-74ba1fffea8f-04_515_707_267_685} A particle \(P\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held with the string taut and making an angle \(\theta\) with the downward vertical. The particle \(P\) is then projected with speed \(\frac { 4 } { 5 } \sqrt { 5 a g }\) perpendicular to the string and just completes a vertical circle (see diagram). Find the value of \(\cos \theta\).

1. A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf { F }\) newtons. The velocity of \(P\) is \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 0\), and \(( 7 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t = 5 \mathrm {~s}\).

Find

1. the speed of \(P\) at \(t = 0\),
2. the vector \(\mathbf { F }\) in the form \(a \mathbf { i } + b \mathbf { j }\),
3. the value of \(t\) when \(P\) is moving parallel to \(\mathbf { i }\).