6.02i Conservation of energy: mechanical energy principle

One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(5mg\), is attached to a fixed point \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The spring hangs with \(P\) vertically below \(A\). The particle \(P\) is released from rest in the position where the extension of the spring is \(\frac{3}{5}a\).

1. Show that the initial acceleration of \(P\) is \(\frac{3}{5}g\) upwards. [3]
2. Find the speed of \(P\) when the spring first returns to its natural length. [4]

\includegraphics{figure_7} One end of a light spring of natural length \(a\) and modulus of elasticity \(4mg\) is attached to a fixed point \(O\). The other end of the spring is attached to a particle \(A\) of mass \(km\), where \(k\) is a constant. Initially the spring lies at rest on a smooth horizontal surface and has length \(a\). A second particle \(B\), of mass \(m\), is moving towards \(A\) with speed \(\sqrt{\frac{4}{3}ga}\) along the line of the spring from the opposite direction to \(O\) (see diagram). The particles \(A\) and \(B\) collide and coalesce. At a point \(C\) in the subsequent motion, the length of the spring is \(\frac{5}{4}a\) and the speed of the combined particle is half of its initial speed.

1. Find the value of \(k\). [6]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(kmg\), is attached to a fixed point A. The other end of the string is attached to a particle \(P\) of mass \(4m\). The particle \(P\) hangs in equilibrium a distance \(x\) vertically below A.

1. Show that \(k = \frac{4a}{x-a}\). [1]

An additional particle, of mass \(2m\), is now attached to \(P\) and the combined particle is released from rest at the original equilibrium position of \(P\). When the combined particle has descended a distance \(\frac{3}{4}a\), its speed is \(\frac{1}{2}\sqrt{ga}\).

1. Find \(x\) in terms of \(a\). [6]

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 attached to a fixed point \(O\). The particle completes vertical circles with centre \(O\). The points A and B are on the path of \(P\), both on the same side of the vertical through \(O\). \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\) and \(OB\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at A is \(u\) and the speed of \(P\) when it is at B is \(\sqrt{ag}\). The tensions in the string at A and B are \(T_A\) and \(T_B\) respectively. It is given that \(T_A = 7T_B\). Find the value of \(\theta\) and find an expression for \(u\) in terms of \(a\) and \(g\). [8]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(kmg\), is attached to a fixed point A. The other end of the string is attached to a particle \(P\) of mass \(4m\). The particle \(P\) hangs in equilibrium a distance \(x\) vertically below A.

1. Show that \(k = \frac{4a}{x-a}\). [1]

An additional particle, of mass \(2m\), is now attached to \(P\) and the combined particle is released from rest at the original equilibrium position of \(P\). When the combined particle has descended a distance \(\frac{3}{4}a\), its speed is \(\frac{1}{3}\sqrt{ga}\).

1. Find \(x\) in terms of \(a\). [6]

One end of a light elastic string of natural length \(0.8\) m and modulus of elasticity \(36\) N is attached to a fixed point \(O\) on a smooth plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{3}{5}\). A particle \(P\) of mass \(2\) kg is attached to the other end of the string. The string lies along a line of greatest slope of the plane with the particle below the level of \(O\). The particle is projected with speed \(\sqrt{2}\) m s\(^{-1}\) directly down the plane from the position where \(OP\) is equal to the natural length of the string. Find the maximum extension of the string during the subsequent motion. [5]

\includegraphics{figure_4} A particle of mass \(m\) 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 is initially held with the string taut at the point \(A\), where \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\). The particle is then projected with speed \(u\) perpendicular to \(OA\) and begins to move upwards in part of a vertical circle. The string goes slack when the particle is at the point \(B\) where angle \(AOB\) is a right angle. The speed of the particle when it is at \(B\) is \(\frac{1}{2}u\) (see diagram). Find the tension in the string at \(A\), giving your answer in terms of \(m\) and \(g\). [8]

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. The particle is held at the point \(A\) with the string taut. The angle between \(OA\) and the downward vertical is equal to \(\alpha\), where \(\cos \alpha = \frac{4}{5}\). The particle is projected from \(A\), perpendicular to the string in an upwards direction, with a speed \(\sqrt{3ga}\). It then moves along a circular path in a vertical plane. The string first goes slack when it makes an angle \(\theta\) with the upward vertical through \(O\). Find the value of \(\cos \theta\). [5]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3mg\), is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The string hangs with \(P\) vertically below \(O\). The particle \(P\) is pulled vertically downwards so that the extension of the string is \(2a\). The particle \(P\) is then released from rest.

1. Find the speed of \(P\) when it is at a distance \(\frac{3}{4}a\) below \(O\). [3]
2. Find the initial acceleration of \(P\) when it is released from rest. [2]

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 attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\), where \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(u\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where angle \(AOB\) is \(90°\) and the speed of \(P\) is \(\sqrt{\frac{1}{3}ag}\).

1. Find the value of \(\sin\theta\). [2]
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is at \(A\). [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 attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\), where \(OA\) makes an angle \(\alpha\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(\sqrt{3ag}\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where \(OB\) makes an angle \(\theta\) with the upward vertical. Given that \(\cos \alpha = \frac{3}{5}\), find the value of \(\cos \theta\). [4]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\lambda mg\), is attached to a fixed point \(O\). The string lies on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected in the direction \(OP\). When the length of the string is \(\frac{4}{3}a\), the speed of \(P\) is \(\sqrt{2ag}\). When the length of the string is \(\frac{5}{3}a\), the speed of \(P\) is \(\frac{1}{2}\sqrt{2ag}\). Find the value of \(\lambda\). [4]

The points \(A\) and \(B\) are at the same horizontal level a distance \(4a\) apart. The ends of a light elastic string, of natural length \(4a\) and modulus of elasticity \(\lambda\), are attached to \(A\) and \(B\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The system is in equilibrium with \(P\) at a distance \(\frac{5}{8}a\) below \(M\), the midpoint of \(AB\).

1. Find \(\lambda\) in terms of \(m\) and \(g\). [3]

The particle \(P\) is pulled down vertically and released from rest at a distance \(\frac{8}{5}a\) below \(M\).

1. Find, in terms of \(a\) and \(g\), the speed of \(P\) as it passes through \(M\) in the subsequent motion. [4]

A smooth sphere with centre \(O\) and of radius \(a\) is fixed to a horizontal plane. A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(u\), so that it begins to move along the surface of the sphere. The particle \(P\) loses contact with the sphere at the point \(Q\) on the sphere, where \(OQ\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\cos\theta = \frac{u^2 + 2ag}{3ag}\). [4]

It is given that \(\cos\theta = \frac{5}{9}\).

1. Find, in terms of \(a\) and \(g\), an expression for the vertical component of the velocity of \(P\) just before it hits the horizontal plane to which the sphere is fixed. [3]
2. Find an expression for the time taken by \(P\) to fall from \(Q\) to the plane. Give your answer in the form \(k\sqrt{\frac{a}{g}}\), stating the value of \(k\) correct to 3 significant figures. [2]

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 attached to a fixed point \(O\). When the particle is hanging vertically below \(O\), it is projected horizontally with speed \(u\) so that it begins to move along a circular path. When \(P\) is at the lowest point of its motion, the tension in the string is \(T\). When \(OP\) makes an angle \(\theta\) with the upward vertical, the tension in the string is \(S\).

1. Show that \(S = T - 3mg(1 + \cos\theta)\). [5]
2. Given that \(u = \sqrt{4ag}\), find the value of \(\cos\theta\) when the string goes slack. [2]

\includegraphics{figure_4} A light spring of natural length \(a\) and modulus of elasticity \(kmg\) is attached to a fixed point \(O\) on a smooth plane inclined to the horizontal at an angle \(\theta\), where \(\sin\theta = \frac{1}{4}\). A particle of mass \(m\) is attached to the lower end of the spring and is held at the point \(A\) on the plane, where \(OA = 2a\) and \(OA\) is along a line of greatest slope of the plane (see diagram). The particle is released from rest and is moving with speed \(V\) when it passes through the point \(B\) on the plane, where \(OB = \frac{3}{2}a\). The speed of the particle is \(\frac{1}{3}V\) when it passes through the point \(C\) on the plane, where \(OC = \frac{3}{4}a\). Find the value of \(k\). [7]

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 \(3mg\), 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. [3]

\includegraphics{figure_2} 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 \(\frac{1}{2}\sqrt{5ag}\) perpendicular to the string and just completes a vertical circle (see diagram). Find the value of \(\cos\theta\). [5]

A light elastic string has natural length \(a\) and modulus of elasticity \(12mg\). One end of the string is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle hangs in equilibrium vertically below \(O\). The particle is pulled vertically down and released from rest with the extension of the string equal to \(e\), where \(e > \frac{1}{4}a\). In the subsequent motion the particle has speed \(\sqrt{2ga}\) when it has ascended a distance \(\frac{1}{4}a\). Find \(e\) in terms of \(a\). [6]

A light spring \(AB\) has natural length \(a\) and modulus of elasticity \(5mg\). The end \(A\) of the spring is attached to a fixed point on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the end \(B\) of the spring. The spring and particle \(P\) are at rest on the surface. Another particle \(Q\) of mass \(km\) is moving with speed \(\sqrt{4ga}\) along the horizontal surface towards \(P\) in the direction \(BA\). The particles \(P\) and \(Q\) collide directly and coalesce. In the subsequent motion the greatest amount by which the spring is compressed is \(\frac{2}{3}a\). Find the value of \(k\). [6]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac{16}{9}Mg\), is attached to a fixed point \(O\). A particle \(P\) of mass \(4M\) is attached to the other end of the string and hangs vertically in equilibrium. Another particle of mass \(2M\) is attached to \(P\) and the combined particle is then released from rest. The speed of the combined particle when it has descended a distance \(\frac{1}{4}a\) is \(v\). Find an expression for \(v\) in terms of \(g\) and \(a\). [6]

\includegraphics{figure_4} A light elastic string has natural length \(8a\) and modulus of elasticity \(5mg\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(AP = BP = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(AB\) it has speed \(\sqrt{80ag}\).

1. Find \(L\) in terms of \(a\). [5]
2. Find the initial acceleration of \(P\) in terms of \(g\). [3]

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible rod of length \(3a\). An identical particle \(Q\) is attached to the other end of the rod. The rod is smoothly pivoted at a point \(O\) on the rod, where \(OQ = x\). The system, of rod and particles, rotates about \(O\) in a vertical plane. At an instant when the rod is vertical, with \(P\) above \(Q\), the particle \(P\) is moving horizontally with speed \(u\). When the rod has turned through an angle of \(60°\) from the vertical, the speed of \(P\) is \(2\sqrt{ag}\), and the tensions in the two parts of the rod, \(OP\) and \(OQ\), have equal magnitudes.

1. Show that the speed of \(Q\) when the rod has turned through an angle of \(60°\) from the vertical is \(\frac{2x}{3a-x}\sqrt{ag}\). [2]
2. Find \(x\) in terms of \(a\). [5]
3. Find \(u\) in terms of \(a\) and \(g\). [4]

\includegraphics{figure_7} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(3a\). The other end of the rod is able to pivot smoothly about the fixed point \(A\). The particle is also attached to one end of a light spring of natural length \(a\) and modulus of elasticity \(kmg\). The other end of the spring is attached to a fixed point \(B\). The points \(A\) and \(B\) are in a horizontal line, a distance \(5a\) apart, and these two points and the rod are in a vertical plane. Initially, \(P\) is held in equilibrium by a vertical force \(F\) with the stretched length of the spring equal to \(4a\) (see diagram). The particle is released from rest in this position and has a speed of \(\frac{6}{5}\sqrt{2ag}\) when the rod becomes horizontal.

1. Find the value of \(k\). [5]
2. Find \(F\) in terms of \(m\) and \(g\). [2]
3. Find, in terms of \(m\) and \(g\), the tension in the rod immediately before it is released. [2]

A particle \(P\) of mass \(m\text{kg}\) is attached to one end of a light elastic string of natural length \(2\text{m}\) and modulus of elasticity \(2mg\text{N}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) hangs in equilibrium vertically below \(O\). The particle \(P\) is pulled down vertically a distance \(d\text{m}\) below its equilibrium position and released from rest.

1. Given that the particle just reaches \(O\) in the subsequent motion, find the value of \(d\). [6]