6.02h Elastic PE: 1/2 k x^2

1 One end of a light elastic string of natural length 1.6 m and modulus of elasticity 25 N is attached to a fixed point \(A\). A particle \(P\) of mass 0.15 kg is attached to the other end of the string. \(P\) is held at rest at a point 2 m vertically below \(A\) and is then released.

1. For the motion from the instant of release until the string becomes slack, find the loss of elastic potential energy and the gain in gravitational potential energy.
2. Hence find the speed of \(P\) at the instant the string becomes slack.

3 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-2_202_972_1619_584} A light elastic string has natural length 0.8 m and modulus of elasticity 12 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level and 0.96 m apart. A particle of weight \(W \mathrm {~N}\) is attached to the mid-point of the string and hangs in equilibrium at a point 0.14 m below \(A B\) (see diagram). Find \(W\).

7 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-4_232_905_762_621} A light elastic string has natural length 10 m and modulus of elasticity 130 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level. A small stone is attached to the mid-point of the string and hangs in equilibrium at a point 2.5 m below \(A B\), as shown in the diagram. With the stone in this position the length of the string is 13 m .

1. Find the tension in the string.
2. Show that the mass of the stone is 3 kg . The stone is now held at rest at a point 8 m vertically below the mid-point of \(A B\).
3. Find the elastic potential energy of the string in this position.
4. The stone is now released. Find the speed with which it passes through the mid-point of \(A B\).

5 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-3_321_698_1692_726} One end of a light elastic string of natural length 4 m and modulus of elasticity 200 N is attached to a fixed point \(A\). The other end is attached to the end \(C\) of a uniform rod \(C D\) of mass 10 kg . One end of another light elastic string, which is identical to the first, is attached to a fixed point \(B\) and the other end is attached to \(D\), as shown in the diagram. The distance \(A B\) is equal to the length of the rod, and \(A B\) is horizontal. The rod is released from rest with \(C\) at \(A\) and \(D\) at \(B\). While the strings are taut, the speed of the rod is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the rod is at a distance of \(( 4 + x ) \mathrm { m }\) below \(A B\).

1. Show that \(v ^ { 2 } = 10 \left( 8 + 2 x - x ^ { 2 } \right)\).
2. Hence find the value of \(x\) when the rod is at its lowest point.

1 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-2_643_218_264_959} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of a light elastic string of natural length 0.8 m and modulus of elasticity 8 N . One end of the string is attached to a fixed point \(A\) and the other end is attached to a fixed point \(B\) which is 2 m vertically below \(A\). When the particle is in equilibrium the distance \(A P\) is 1.1 m (see diagram). Find the value of \(m\).

4 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. \(P\) is released from rest at a point on the table 3.5 m from \(O\). The speed of \(P\) at the instant the string becomes slack is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the work done against friction during the period from the release of \(P\) until the string becomes slack,
2. the coefficient of friction between \(P\) and the table.

1 \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-2_248_1267_276_440} A light elastic string has natural length 0.6 m and modulus of elasticity \(\lambda \mathrm { N }\). The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level and 0.63 m apart. A particle \(P\) of mass 0.064 kg is attached to the mid-point of the string and hangs in equilibrium at a point 0.08 m below \(A B\) (see diagram). Find

1. the tension in the string,
2. the value of \(\lambda\).

6 A light elastic string has natural length 2 m and modulus of elasticity 0.8 N . One end of the string is attached to a fixed point \(O\) of a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 12 } { 13 }\). A particle \(P\) of mass 0.052 kg is attached to the other end of the string. The coefficient of friction between the particle and the plane is 0.4 . \(P\) is released from rest at \(O\).

1. When \(P\) has moved \(d\) metres down the plane from \(O\), where \(d > 2\), find expressions in terms of \(d\) for
   1. the loss in gravitational potential energy of \(P\),
   2. the gain in elastic potential energy of the string,
   3. the work done by the frictional force acting on \(P\).
   4. Show that \(d ^ { 2 } - 6 d + 4 = 0\) when \(P\) is at its lowest point, and hence find the value of \(d\) in this case.

5 One end of a light elastic string, of natural length 0.5 m and modulus of elasticity 140 N , is attached to a fixed point \(O\). A particle of mass 0.8 kg is attached to the other end of the string. The particle is released from rest at \(O\). By considering the energy of the system, find

1. the speed of the particle when the extension of the string is 0.1 m ,
2. the extension of the string when the particle is at its lowest point.

6 \includegraphics[max width=\textwidth, alt={}, center]{57f7ca89-f028-447a-9ac9-55f931201e6b-3_83_771_1978_689} \(A\) and \(B\) are fixed points on a smooth horizontal table. The distance \(A B\) is 2.5 m . An elastic string of natural length 0.6 m and modulus of elasticity 24 N has one end attached to the table at \(A\), and the other end attached to a particle \(P\) of mass 0.95 kg . Another elastic string of natural length 0.9 m and modulus of elasticity 18 N has one end attached to the table at \(B\), and the other end attached to \(P\). The particle \(P\) is held at rest at the mid-point of \(A B\) (see diagram).

1. Find the tensions in the strings. The particle is released from rest.
2. Find the acceleration of \(P\) immediately after its release.
3. \(P\) reaches its maximum speed at the point \(C\). Find the distance \(A C\).

1 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-2_549_775_269_685} A particle \(A\) and a block \(B\) are attached to opposite ends of a light elastic string of natural length 2 m and modulus of elasticity 6 N . The block is at rest on a rough horizontal table. The string passes over a small smooth pulley \(P\) at the edge of the table, with the part \(B P\) of the string horizontal and of length 1.2 m . The frictional force acting on \(B\) is 1.5 N and the system is in equilibrium (see diagram). Find the distance \(P A\).

6 One end of a light elastic string of natural length 1.25 m and modulus of elasticity 20 N is attached to a fixed point \(O\). A particle \(P\) of mass 0.5 kg is attached to the other end of the string. \(P\) is held at rest at \(O\) and then released. When the extension of the string is \(x \mathrm {~m}\) the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = - 32 x ^ { 2 } + 20 x + 25\).
2. Find the maximum speed of \(P\).
3. Find the acceleration of \(P\) when it is at its lowest point.

6 \includegraphics[max width=\textwidth, alt={}, center]{fb79f949-567c-4dbb-8533-7b7278cad21c-3_200_639_1754_753} A particle \(P\) of mass 1.6 kg is attached to one end of each of two light elastic strings. The other ends of the strings are attached to fixed points \(A\) and \(B\) which are 2 m apart on a smooth horizontal table. The string attached to \(A\) has natural length 0.25 m and modulus of elasticity 4 N , and the string attached to \(B\) has natural length 0.25 m and modulus of elasticity 8 N . The particle is held at the mid-point \(M\) of \(A B\) (see diagram).

1. Find the tensions in the strings.
2. Show that the total elastic potential energy in the two strings is 13.5 J . \(P\) is released from rest and in the subsequent motion both strings remain taut. The displacement of \(P\) from \(M\) is denoted by \(x \mathrm {~m}\). Find
3. the initial acceleration of \(P\),
4. the non-zero value of \(x\) at which the speed of \(P\) is zero. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fb79f949-567c-4dbb-8533-7b7278cad21c-4_529_542_269_804} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A uniform solid body has a cross-section as shown in Fig. 1.
5. Show that the centre of mass of the body is 2.5 cm from the plane face containing \(O B\) and 3.5 cm from the plane face containing \(O A\).
6. The solid is placed on a rough plane which is initially horizontal. The coefficient of friction between the solid and the plane is \(\mu\).
   1. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{fb79f949-567c-4dbb-8533-7b7278cad21c-4_332_469_1320_918} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure} The solid is placed with \(O A\) in contact with the plane, and then the plane is tilted so that \(O A\) lies along a line of greatest slope with \(A\) higher than \(O\) (see Fig. 2). When the angle of inclination is sufficiently great the solid starts to topple (without sliding). Show that \(\mu > \frac { 5 } { 7 }\).
      [0pt] [5]
   2. \includegraphics[max width=\textwidth, alt={}, center]{fb79f949-567c-4dbb-8533-7b7278cad21c-4_291_465_1987_918} Instead, the solid is placed with \(O B\) in contact with the plane, and then the plane is tilted so that \(O B\) lies along a line of greatest slope with \(B\) higher than \(O\) (see Fig. 3). When the angle of inclination is sufficiently great the solid starts to slide (without toppling). Find another inequality for \(\mu\).

6 \includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-4_324_1267_794_440} A particle \(P\) of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m . The ends of the string are attached to fixed points \(A\) and \(B\) which are 4.8 m apart at the same horizontal level. \(P\) hangs in equilibrium at a point 0.7 m vertically below the mid-point \(M\) of \(A B\) (see diagram).

1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N . \(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.
2. Find the speed with which \(P\) passes through \(M\).

7 One end of a light elastic string of natural length 3 m and modulus of elasticity 24 N is attached to a fixed point \(O\). A particle \(P\) of mass 0.4 kg is attached to the other end of the string. \(P\) is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the extension of the string is \(x \mathrm {~m}\) the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 64 + 20 x - 20 x ^ { 2 }\).
2. Find the greatest speed of the particle.
3. Calculate the greatest tension in the string.

3 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_247_839_1375_653} A light elastic string of natural length 1.2 m and modulus of elasticity 24 N is attached to fixed points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 1.2 \mathrm {~m}\). A particle \(P\) is attached to the mid-point of the string. \(P\) is projected with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the surface in a direction perpendicular to \(A B\) (see diagram). \(P\) comes to instantaneous rest at a distance 0.25 m from \(A B\).

1. Show that the mass of \(P\) is 0.8 kg .
2. Calculate the greatest deceleration of \(P\).

5 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-3_348_1205_251_470} \(A B C\) is a uniform triangular lamina of weight 19 N , with \(A B = 0.22 \mathrm {~m}\) and \(A C = B C = 0.61 \mathrm {~m}\). The plane of the lamina is vertical. \(A\) rests on a rough horizontal surface, and \(A B\) is vertical. The equilibrium of the lamina is maintained by a light elastic string of natural length 0.7 m which passes over a small smooth peg \(P\) and is attached to \(B\) and \(C\). The portion of the string attached to \(B\) is horizontal, and the portion of the string attached to \(C\) is vertical (see diagram).

1. Show that the tension in the string is 10 N .
2. Calculate the modulus of elasticity of the string.
3. Find the magnitude and direction of the force exerted by the surface on the lamina at \(A\).

4 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 12 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.24 \mathrm {~kg} . P\) is projected vertically upwards with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a position 0.8 m vertically below \(O\).

1. Calculate the speed of the particle when it is moving upwards with zero acceleration.
2. Show that the particle moves 0.6 m while it is moving upwards with constant acceleration.

4 The ends of a light elastic string of natural length 0.8 m and modulus of elasticity \(\lambda \mathrm { N }\) are attached to fixed points \(A\) and \(B\) which are 1.2 m apart at the same horizontal level. A particle of mass 0.3 kg is attached to the centre of the string, and released from rest at the mid-point of \(A B\). The particle descends 0.32 m vertically before coming to instantaneous rest.

1. Calculate \(\lambda\).
2. Calculate the speed of the particle when it is 0.25 m below \(A B\).

5 One end of a light elastic string of natural length 0.3 m and modulus of elasticity 6 N is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass 0.2 kg , which moves on the plane in a circular path with centre \(O\). The angular speed of \(P\) is \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. For the case \(\omega = 5\), calculate the extension of the string.
2. Express the extension of the string in terms of \(\omega\), and hence find the set of possible value of \(\omega\).

4 A light elastic string has natural length 2.4 m and modulus of elasticity 21 N . A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which are 2.4 m apart at the same horizontal level. \(P\) is projected vertically upwards with velocity \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the mid-point of \(A B\). In the subsequent motion \(P\) is at instantaneous rest at a point 1.6 m above \(A B\).

1. Find \(m\).
2. Calculate the acceleration of \(P\) when it first passes through a point 0.5 m below \(A B\).

3 A light elastic string has natural length 2.2 m and modulus of elasticity 14.3 N . A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which are 2.4 m apart at the same horizontal level. \(P\) is released from rest at the mid-point of \(A B\). In the subsequent motion \(P\) has its greatest speed at a point 0.5 m below \(A B\).

1. Find \(m\).
2. Calculate the greatest speed of \(P\).

4 \includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_170_616_1649_767} A small sphere \(S\) of mass \(m \mathrm {~kg}\) is moving inside a fixed smooth hollow cylinder whose axis is vertical. \(S\) moves with constant speed in a horizontal circle of radius 0.4 m and is in contact with both the plane base and the curved surface of the cylinder (see diagram).

1. Given that the horizontal and vertical forces exerted on \(S\) by the cylinder have equal magnitudes, calculate the speed of \(S\). \(S\) is now attached to the centre of the base of the cylinder by a horizontal light elastic string of natural length 0.25 m and modulus of elasticity 13 N . The sphere \(S\) is set in motion and moves in a horizontal circle with constant angular speed \(\omega \mathrm { rads } ^ { - 1 }\) and is in contact with both the plane base and the curved surface of the cylinder.
2. It is given that the magnitudes of the horizontal and vertical forces exerted on \(S\) by the cylinder are equal if \(\omega = 8\). Calculate \(m\).
3. For the value of \(m\) found in part (ii), find the least possible value of \(\omega\) for the motion.

5 A light elastic string has natural length 3 m and modulus of elasticity 45 N . A particle \(P\) of mass 0.6 kg is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie on a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is 4 m , and \(A\) is higher than \(B\).

1. Calculate the distance \(A P\) when \(P\) rests on the slope in equilibrium. \(P\) is released from rest at the point between \(A\) and \(B\) where \(A P = 2.5 \mathrm {~m}\).
2. Find the maximum speed of \(P\).
3. Show that \(P\) is at rest when \(A P = 1.6 \mathrm {~m}\).

2 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 19.2 N . The other end of the string is attached to a fixed point \(A\). The particle \(P\) is released from rest at the point 2.7 m vertically above \(A\). Calculate

1. the initial acceleration of \(P\),
2. the speed of \(P\) when it reaches \(A\).