Elastic string on rough inclined plane

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.

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point \(O\) on a rough plane which is inclined to the horizontal at an angle \(\alpha\) The string lies along a line of greatest slope of the plane.
The particle \(P\) is held at rest on the plane at the point \(A\), where \(O A = a\), as shown in Figure 5. The particle \(P\) is released from \(A\) and slides down the plane, coming to rest at the point \(B\). The coefficient of friction between \(P\) and the plane is \(\mu\), where \(\mu < \tan \alpha\) Air resistance is modelled as being negligible.

1. Show that \(A B = a ( \sin \alpha - \mu \cos \alpha )\). Given that \(\tan \alpha = \frac { 3 } { 4 }\) and \(\mu = \frac { 1 } { 2 }\)
2. find, in terms of \(a\) and \(g\), the maximum speed of \(P\) as it moves from \(A\) to \(B\)
3. Describe the motion of \(P\) after it reaches \(B\), justifying your answer.

4. \begin{figure}[h] \end{figure} Figure 2 shows a light elastic string, of modulus of elasticity \(\lambda\) newtons and natural length 0.6 m . One end of the string is attached to a fixed point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass 0.5 kg . The string lies along a line of greatest slope of the plane. The particle is held at rest on the plane at the point \(B\), where \(B\) is lower than \(A\) and \(A B = 1.2 \mathrm {~m}\). The particle then receives an impulse of magnitude 1.5 N s in the direction parallel to the string, causing \(P\) to move up the plane towards \(A\). The coefficient of friction between \(P\) and the plane is 0.7 . Given that \(P\) comes to rest at the instant when the string becomes slack, find the value of \(\lambda\).

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 12 N . The other end of the spring is attached to a fixed point \(A\) on a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. Initially \(P\) is held at rest on the plane at the point \(B\), where \(B\) is below \(A\), with \(A B = 0.3 \mathrm {~m}\) and \(A B\) lies along a line of greatest slope of the plane. The point \(C\) lies on the plane with \(A C = 1 \mathrm {~m}\), as shown in Figure 3. The coefficient of friction between \(P\) and the plane is 0.3 After being released \(P\) passes through the point \(C\). Find the speed of \(P\) at the instant it passes through \(C\).

3. \begin{figure}[h] \end{figure} 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 \(\lambda\).

4. \begin{figure}[h] \end{figure} One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(3 m g\), is fixed to a point \(A\) on a fixed plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\) A small ball of mass \(2 m\) is attached to the free end of the string. The ball is held at a point \(C\) on the plane, where \(C\) is below \(A\) and \(A C = l\) as shown in Figure 3. The string is parallel to a line of greatest slope of the plane. The ball is released from rest. In an initial model the plane is assumed to be smooth.

1. Find the distance that the ball moves before first coming to instantaneous rest. In a refined model the plane is assumed to be rough. The coefficient of friction between the ball and the plane is \(\mu\). The ball first comes to instantaneous rest after moving a distance \(\frac { 2 } { 5 } l\).
2. Find the value of \(\mu\).

4. One end of a light elastic string, of modulus of elasticity \(2 m g\) and natural length \(l\), is fixed to a point \(O\) on a rough plane. The plane is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The other end of the string is attached to a particle \(P\) of mass \(m\) which is held at rest on the plane at the point \(O\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The particle is released from rest and slides down the plane, coming to instantaneous rest at the point \(A\), where \(O A = k l\). Given that \(k > 1\), find, to 3 significant figures, the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-13_152_72_118_127} \includegraphics[max width=\textwidth, alt={}]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-13_90_1620_123_203} □ ⟶ \(\_\_\_\_\) T

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity \(\lambda\). The other end of the string is fixed to a point \(A\) on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 6 } \sqrt { 3 }\). \(P\) is held at rest at \(A\) and then released. It first comes to instantaneous rest at the point \(B , 2.2 \mathrm {~m}\) from \(A\). For the motion of \(P\) from \(A\) to \(B\),

1. show that the work done against friction is 10.78 J ,
2. find the change in the gravitational potential energy of \(P\). By using the work-energy principle, or otherwise,
3. find \(\lambda\).

9 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A light elastic string has one end attached to a fixed point, \(A\), on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle, \(P\), of mass 2 kg .
The elastic string has natural length 1.3 metres and modulus of elasticity 65 N .
The particle is pulled down the plane in the direction of the line of greatest slope through \(A\).
The particle is released from rest when it is 2 metres from \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-14_549_744_861_785} The coefficient of friction between the particle and the plane is 0.6
After the particle is released it moves up the plane.
The particle comes to rest at a point \(B\), which is a distance, \(d\) metres, from \(A\). 9

1. Show that the value of \(d\) is 1.4.
   [0pt] [7 marks] 9
2. Determine what happens after \(P\) reaches the point \(B\). Fully justify your answer.
   [0pt] [3 marks]

1. A light elastic string with natural length \(l\) and modulus of elasticity \(k m g\) has one end attached to a fixed point \(A\) on a rough inclined plane. The other end of the string is attached to a package of mass \(m\).

The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) The package is initially held at \(A\). The package is then projected with speed \(\sqrt { 6 g l }\) up a line of greatest slope of the plane and first comes to rest at the point \(B\), where \(A B = 31\).
The coefficient of friction between the package and the plane is \(\frac { 1 } { 4 }\) By modelling the package as a particle,

1. show that \(k = \frac { 15 } { 26 }\)
2. find the acceleration of the package at the instant it starts to move back down the plane from the point \(B\).

6. \begin{figure}[h] \end{figure} A light elastic spring has natural length \(3 l\) and modulus of elasticity \(3 m g\).
One end of the spring is attached to a fixed point \(X\) on a rough inclined plane.
The other end of the spring is attached to a package \(P\) of mass \(m\).
The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\) The package is initially held at the point \(Y\) on the plane, where \(X Y = l\). The point \(Y\) is higher than \(X\) and \(X Y\) is a line of greatest slope of the plane, as shown in Figure 2. The package is released from rest at \(Y\) and moves up the plane.
The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 3 }\) By modelling \(P\) as a particle,

1. show that the acceleration of \(P\) at the instant when \(P\) is released from rest is \(\frac { 17 } { 15 } \mathrm {~g}\)
2. find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the spring first reaches its natural length of 31 .

\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]

An elastic string has natural length 1.5 metres and modulus of elasticity 120 newtons. One end of the string is attached to a fixed point, \(A\), on a rough plane inclined at \(20°\) to the horizontal. The other end of the elastic string is attached to a particle of mass 4 kg. The coefficient of friction between the particle and the plane is 0.8. The three points, \(A\), \(B\) and \(C\), lie on a line of greatest slope. The point \(C\) is \(x\) metres from \(A\), as shown in the diagram. The particle is released from rest at \(C\) and moves up the plane. \includegraphics{figure_8}

1. Show that, as the particle moves up the plane, the frictional force acting on the particle is 29.5 N, correct to three significant figures. [3 marks]
2. The particle comes to rest for an instant at \(B\), which is 2 metres from \(A\). The particle then starts to move back towards \(A\).
   1. Find \(x\). [8 marks]
   2. Find the acceleration of the particle as it starts to move back towards \(A\). [4 marks]

A particle \(P\), of mass \(5\) kg is placed at the point \(A\) on a rough plane which is inclined at \(30°\) to the horizontal. The points \(Q\) and \(R\) are also on the surface of the inclined plane, with \(QR = 15\) metres. The point \(A\) is between \(Q\) and \(R\) so that \(AQ = 4\) metres and \(AR = 11\) metres. The three points \(Q\), \(A\) and \(R\) are on a line of greatest slope of the plane. \includegraphics{figure_8} The particle is attached to two light elastic strings, \(PQ\) and \(PR\). One of the strings, \(PQ\), has natural length \(4\) metres and modulus of elasticity \(160\) N, the other string, \(PR\), has natural length \(6\) metres and modulus of elasticity \(120\) N. The particle is released from rest at the point \(A\). The coefficient of friction between the particle and the plane is \(0.4\). Find the distance of the particle from \(Q\) when it is next at rest. [8 marks]

\includegraphics{figure_4} A light elastic spring has natural length \(l\) and modulus of elasticity \(4mg\). One end of the spring is attached to a point \(A\) on a plane that is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac{1}{4}\). The particle \(P\) is held at a point \(B\) on the plane where \(B\) is below \(A\) and \(AB = l\), with the spring lying along a line of greatest slope of the plane, as shown in Figure 4. At time \(t = 0\), the particle is projected up the plane towards \(A\) with speed \(\frac{1}{2}\sqrt{gl}\). At time \(t\), the compression of the spring is \(x\).

1. Show that $$\frac{d^2x}{dt^2} + 4\omega^2x = -g, \text{ where } \omega = \sqrt{\frac{g}{l}}.$$ [6]

1. Find \(x\) in terms of \(l\), \(\omega\) and \(t\). [7]

1. Find the distance that \(P\) travels up the plane before first coming to rest. [4]

One end of a light elastic string of natural length \(2.1\) m and modulus of elasticity \(4.8\) N is attached to a particle, \(P\), of mass \(1.75\) kg. The other end of the string is attached to a fixed point, \(O\), which is on a rough inclined plane. The angle between the plane and the horizontal is \(\theta\) where \(\sin\theta = \frac{3}{5}\). The coefficient of friction between \(P\) and the plane is \(0.732\). Particle \(P\) is placed on the plane at \(O\) and then projected down a line of greatest slope of the plane with an initial speed of \(2.4\) m s\(^{-1}\). Determine the distance that \(P\) has travelled from \(O\) at the instant when it first comes to rest. You can assume that during its motion \(P\) does not reach the bottom of the inclined plane. [8]