6.02i Conservation of energy: mechanical energy principle

A particle \(P\) of mass \(m\) is fired vertically upwards from a point on the surface of the Earth and initially moves in a straight line directly away from the centre of the Earth. When \(P\) is at a distance \(x\) from the centre of the Earth, the gravitational force exerted by the Earth on \(P\) is directed towards the centre of the Earth and has a magnitude which is inversely proportional to \(x^2\). At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a fixed sphere of radius \(R\).

1. Show that the magnitude of the gravitational force acting on \(P\) is \(\frac{mgR^2}{x^2}\) [2]

The particle was fired with initial speed \(U\) and the greatest height above the surface of the Earth reached by \(P\) is \(\frac{R}{20}\). Given that air resistance can be ignored,

1. find \(U\) in terms of \(g\) and \(R\). [7]

\includegraphics{figure_2} A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity \(\lambda\) newtons. The other end of the spring is attached to a fixed point \(O\) on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). The coefficient of friction between the particle and the plane is 0.15. The particle is held on the plane at a point which is 1.5 m down the line of greatest slope from \(O\), as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of \(\lambda\). [9]

\includegraphics{figure_5} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is fixed at the point \(O\). The particle is initially held with \(OP\) horizontal and the string taut. It is then projected vertically upwards with speed \(u\), where \(u^2 = 5ag\). When \(OP\) has turned through an angle \(\theta\) the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 5.

1. Find, in terms of \(a\), \(g\) and \(\theta\), an expression for \(v^2\). [3]
2. Find, in terms of \(m\), \(g\) and \(\theta\), an expression for \(T\). [4]
3. Prove that \(P\) moves in a complete circle. [3]
4. Find the maximum speed of \(P\). [2]

A light elastic string, of natural length \(3a\) and modulus of elasticity \(6mg\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(2m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\), vertically below \(A\).

1. Find the distance \(AO\). [3]

The particle is now raised to point \(C\) vertically below \(A\), where \(AC > 3a\), and is released from rest.

1. Show that \(P\) moves with simple harmonic motion of period \(2\pi\sqrt{\frac{a}{g}}\). [5]

It is given that \(OC = \frac{1}{4}a\).

1. Find the greatest speed of \(P\) during the motion. [3]

The point \(D\) is vertically above \(O\) and \(OD = \frac{1}{8}a\). The string is cut as \(P\) passes through \(D\), moving upwards.

1. Find the greatest height of \(P\) above \(O\) in the subsequent motion. [4]

One end of a light elastic string, of natural length 2 m and modulus of elasticity 19.6 N, is attached to a fixed point \(A\). A small ball \(B\) of mass 0.5 kg is attached to the other end of the string. The ball is released from rest at \(A\) and first comes to instantaneous rest at the point \(C\), vertically below \(A\).

1. Find the distance \(AC\). [6]
2. Find the instantaneous acceleration of \(B\) at \(C\). [3]

\includegraphics{figure_3} Figure 3 shows a fixed hollow sphere of internal radius \(a\) and centre \(O\). A particle \(P\) of mass \(m\) is projected horizontally from the lowest point \(A\) of a sphere with speed \(\sqrt{\left(\frac{5}{4}ag\right)}\). It moves in a vertical circle, centre \(O\), on the smooth inner surface of the sphere. The particle passes through the point \(B\), which is in the same horizontal plane as \(O\). It leaves the surface of the sphere at the point \(C\), where \(OC\) makes an angle \(\theta\) with the upward vertical.

1. Find, in terms of \(m\) and \(g\), the normal reaction between \(P\) and the surface of the sphere at \(B\). [4]
2. Show that \(\theta = 60°\). [7]

After leaving the surface of the sphere, \(P\) meets it again at the point \(A\).

1. Find, in terms of \(a\) and \(g\), the time \(P\) takes to travel from \(C\) to \(A\). [4]

A light spring of natural length \(L\) has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The particle is moving vertically. As it passes through the point \(B\) below \(A\), where \(AB = L\), its speed is \(\sqrt{(2gL)}\). The particle comes to instantaneous rest at a point \(C\), \(4L\) below \(A\).

1. Show that the modulus of elasticity of the spring is \(\frac{8mg}{9}\). [4]

At the point \(D\) the tension in the spring is \(mg\).

1. Show that \(P\) performs simple harmonic motion with centre \(D\). [5]
2. Find, in terms of \(L\) and \(g\),
   1. the period of the simple harmonic motion,
   2. the maximum speed of \(P\).
   [5]

\includegraphics{figure_6} A trapeze artiste of mass 60 kg is attached to the end \(A\) of a light inextensible rope \(OA\) of length 5 m. The artiste must swing in an arc of a vertical circle, centre \(O\), from a platform \(P\) to another platform \(Q\), where \(PQ\) is horizontal. The other end of the rope is attached to the fixed point \(O\) which lies in the vertical plane containing \(PQ\), with \(\angle POQ = 120^{\circ}\) and \(OP = OQ = 5\) m, as shown in Figure 6. As part of her act, the artiste projects herself from \(P\) with speed \(\sqrt{15}\) m s\(^{-1}\) in a direction perpendicular to the rope \(OA\) and in the plane \(POQ\). She moves in a circular arc towards \(Q\). At the lowest point of her path she catches a ball of mass \(m\) kg which is travelling towards her with speed 3 m s\(^{-1}\) and parallel to \(QP\). After catching the ball, she comes to rest at the point \(Q\). By modelling the artiste and the ball as particles and ignoring her air resistance, find

1. the speed of the artiste immediately before she catches the ball, [4]
2. the value of \(m\), [7]
3. the tension in the rope immediately after she catches the ball. [3]

\includegraphics{figure_4} A small ball of mass \(3m\) is attached to the ends of two light elastic strings \(AP\) and \(BP\), each of natural length \(l\) and modulus of elasticity \(kmg\). The ends \(A\) and \(B\) of the strings are attached to fixed points on the same horizontal level, with \(AB = 2l\). The mid-point of \(AB\) is \(C\). The ball hangs in equilibrium at a distance \(\frac{3}{4}l\) vertically below \(C\) as shown in Figure 4.

1. Show that \(k = 10\) [7]

The ball is now pulled vertically downwards until it is at a distance \(\frac{15}{8}l\) below \(C\). The ball is released from rest.

1. Find the speed of the ball as it reaches \(C\). [6]

\includegraphics{figure_5} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can turn freely in a vertical plane about \(O\). The particle is projected with speed \(u\) from a point \(A\), where \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\) and \(0 < \alpha < \frac{\pi}{2}\). When \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\) the speed of \(P\) is \(v\) as shown in Figure 5.

1. Show that \(v^2 = u^2 + 2gl (\cos \alpha - \cos \theta)\). [4]

It is given that \(\cos \alpha = \frac{3}{5}\) and that \(P\) moves in a complete vertical circle.

1. Show that \(u > 2\sqrt{\frac{gl}{5}}\). [4]

As the rod rotates the least tension in the rod is \(T\) and the greatest tension is \(5T\).

1. Show that \(u^2 = \frac{33}{10}gl\). [9]

\includegraphics{figure_1} A smooth solid hemisphere, of radius 0.8 m and centre \(O\), is fixed with its plane face on a horizontal table. A particle of mass 0.5 kg is projected horizontally with speed \(u\) m s\(^{-1}\) from the highest point \(A\) of the hemisphere. The particle leaves the hemisphere at the point \(B\), which is a vertical distance of 0.2 m below the level of \(A\). The speed of the particle at \(B\) is \(v\) m s\(^{-1}\) and the angle between \(OA\) and \(OB\) is \(\theta\), as shown in Fig. 1.

1. Find the value of \(\cos \theta\). [1]
2. Show that \(v^2 = 5.88\). [3]
3. Find the value of \(u\). [3]

\includegraphics{figure_2} A light horizontal spring, of natural length 0.25 m and modulus of elasticity 52 N, is fastened at one end to a point \(A\). The other end of the spring is fastened to a small wooden block \(B\) of mass 1.5 kg which is on a horizontal table, as shown in Fig. 2. The block is modelled as a particle. The table is initially assumed to be smooth. The block is released from rest when it is a distance 0.3 m from \(A\). By using the principle of the conservation of energy,

1. find, to 3 significant figures, the speed of \(B\) when it is a distance 0.25 m from \(A\). [5]

It is now assumed that the table is rough and the coefficient of friction between \(B\) and the table is 0.6.

1. Find, to 3 significant figures, the minimum distance from \(A\) at which \(B\) can rest in equilibrium. [5]

\includegraphics{figure_5} A small ring \(R\) of mass \(m\) is free to slide on a smooth straight wire which is fixed at an angle of \(30°\) to the horizontal. The ring is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point \(B\), where \(AB = \frac{a}{2}\).

1. Show that \(\lambda = 4mg\). [3]

The ring is pulled down to the point \(C\), where \(BC = \frac{1}{4}a\), and released from rest. At time \(t\) after \(R\) is released the extension of the string is \((\frac{1}{4}a + x)\).

1. Obtain a differential equation for the motion of \(R\) while the string remains taut, and show that it represents simple harmonic motion with period \(\pi\sqrt{\left(\frac{a}{g}\right)}\). [6]
2. Find, in terms of \(g\), the greatest magnitude of the acceleration of \(R\) while the string remains taut. [2]
3. Find, in terms of \(a\) and \(g\), the time taken for \(R\) to move from the point at which it first reaches maximum speed to the point where the string becomes slack for the first time. [5]

A light elastic string \(AB\) of natural length 1.5 m has modulus of elasticity 20 N. The end \(A\) is fixed to a point on a smooth horizontal table. A small ball \(S\) of mass 0.2 kg is attached to the end \(B\). Initially \(S\) is at rest on the table with \(AB = 1.5\) m. The ball \(S\) is then projected horizontally directly away from \(A\) with a speed of 5 m s\(^{-1}\). By modelling \(S\) as a particle,

1. find the speed of \(S\) when \(AS = 2\) m. [5]

When the speed of \(S\) is 1.5 m s\(^{-1}\), the string breaks.

1. Find the tension in the string immediately before the string breaks. [5]

A light elastic string, of natural length \(4a\) and modulus of elasticity \(8mg\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\).

1. Find the distance \(AO\). [2]

The particle is now pulled down to a point \(C\) vertically below \(O\), where \(OC = d\). It is released from rest. In the subsequent motion the string does not become slack.

1. Show that \(P\) moves with simple harmonic motion of period \(\pi\sqrt{\frac{2a}{g}}\). [7]

The greatest speed of \(P\) during this motion is \(\frac{1}{2}\sqrt{(ga)}\).

1. Find \(d\) in terms of \(a\). [3]

Instead of being pulled down a distance \(d\), the particle is pulled down a distance \(a\). Without further calculation,

1. describe briefly the subsequent motion of \(P\). [2]

\includegraphics{figure_3} A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is hanging at the point \(A\), which is vertically below \(O\). It is projected horizontally with speed \(u\). When the particle is at the point \(P\), \(\angle AOP = \theta\), as shown in Fig. 3. The string oscillates through an angle \(\alpha\) on either side of \(OA\) where \(\cos \alpha = \frac{2}{3}\).

1. Find \(u\) in terms of \(g\) and \(l\). [4]

When \(\angle AOP = \theta\), the tension in the string is \(T\).

1. Show that \(T = \frac{mg}{3}(9\cos\theta - 4)\). [6]
2. Find the range of values of \(T\). [4]

A particle \(P\) of mass \(m\) is held at a point \(A\) on a rough horizontal plane. The coefficient of friction between \(P\) and the plane is \(\frac{2}{3}\). The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4mg\). The other end of the string is attached to a fixed point \(O\) on the plane, where \(OA = \frac{3}{4}a\). The particle \(P\) is released from rest and comes to rest at a point \(B\), where \(OB < a\). Using the work-energy principle, or otherwise, calculate the distance \(AB\). [6]

\includegraphics{figure_2} A particle is at the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(u\), where \(u < \sqrt{ag}\). The particle leaves the sphere at the point \(C\), where \(OC\) makes an angle \(\theta\) with the upward vertical, as shown in Fig. 2.

1. Find an expression for \(\cos \theta\) in terms of \(u\), \(g\) and \(a\). [7]

The particle strikes the plane with speed \(\sqrt{\frac{9ag}{2}}\).

1. Find, to the nearest degree, the value of \(\theta\). [7]

Two light elastic strings each have natural length \(0.75\) m and modulus of elasticity \(49\) N. A particle \(P\) of mass \(2\) kg is attached to one end of each string. The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(AB\) is horizontal and \(AB = 1.5\) m. \includegraphics{figure_2} The particle is held at the mid-point of \(AB\). The particle is released from rest, as shown in Figure 2.

1. Find the speed of \(P\) when it has fallen a distance of \(1\) m. [6]

Given instead that \(P\) hangs in equilibrium vertically below the mid-point of \(AB\), with \(\angle APB = 2\alpha\),

1. show that \(\tan \alpha + 5 \sin \alpha = 5\). [6]

One end of a light inextensible string of length \(l\) is attached to a particle \(P\) of mass \(m\). The other end is attached to a fixed point \(A\). The particle is hanging freely at rest with the string vertical when it is projected horizontally with speed \(\sqrt{\frac{5gl}{2}}\).

1. Find the speed of \(P\) when the string is horizontal. [4]

When the string is horizontal it comes into contact with a small smooth fixed peg which is at the point \(B\), where \(AB\) is horizontal, and \(AB < l\). Given that the particle then describes a complete semicircle with centre \(B\),

1. Find the least possible value of the length \(AB\). [9]

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_1} A light elastic string, of natural length \(3l\) and modulus of elasticity \(\lambda\), has its ends attached to two points \(A\) and \(B\), where \(AB = 3l\) and \(AB\) is horizontal. A particle \(P\) of mass \(m\) is attached to the mid-point of the string. Given that \(P\) rests in equilibrium at a distance \(2l\) below \(AB\), as shown in Figure 1,

1. show that \(\lambda = \frac{15mg}{16}\) [9]

The particle is pulled vertically downwards from its equilibrium position until the total length of the elastic string is \(7.8l\). The particle is released from rest.

1. Show that \(P\) comes to instantaneous rest on the line \(AB\). [6]

A light elastic string has natural length \(8\) m and modulus of elasticity \(80\) N. The ends of the string are attached to fixed points \(P\) and \(Q\) which are on the same horizontal level and \(12\) m apart. A particle is attached to the mid-point of the string and hangs in equilibrium at a point \(4.5\) m below \(PQ\).

1. Calculate the weight of the particle. [6]
2. Calculate the elastic energy in the string when the particle is in this position. [3]

One end of a light inextensible string of length \(l\) is attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\), which is held at a point \(B\) with the string taut and \(AP\) making an angle arccos \(\frac{1}{4}\) with the downward vertical. The particle is released from rest. When \(AP\) makes an angle \(\theta\) with the downward vertical, the string is taut and the tension in the string is \(T\).

1. Show that $$T = 3mg \cos \theta - \frac{mg}{2}.$$ [6]

\includegraphics{figure_3} At an instant when \(AP\) makes an angle of \(60°\) to the downward vertical, \(P\) is moving upwards, as shown in Figure 3. At this instant the string breaks. At the highest point reached in the subsequent motion, \(P\) is at a distance \(d\) below the horizontal through \(A\).

1. Find \(d\) in terms of \(l\). [5]

\includegraphics{figure_4} \(A\) and \(B\) are two points on a smooth horizontal floor, where \(AB = 5\) m. A particle \(P\) has mass \(0.5\) kg. One end of a light elastic spring, of natural length \(2\) m and modulus of elasticity \(16\) N, is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length \(1\) m and modulus of elasticity \(12\) N, are attached to \(P\) and \(B\), as shown in Figure 4.

1. Find the extensions in the two springs when the particle is at rest in equilibrium. [5]

Initially \(P\) is at rest in equilibrium. It is then set in motion and starts to move towards \(B\). In the subsequent motion \(P\) does not reach \(A\) or \(B\).

1. Show that \(P\) oscillates with simple harmonic motion about the equilibrium position. [4]
2. Given that the initial speed of \(P\) is \(\sqrt{10}\) m s\(^{-1}\), find the proportion of time in each complete oscillation for which \(P\) stays within \(0.25\) m of the equilibrium position. [7]