Edexcel M3 (Mechanics 3) 2020 June

1. \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-03_62_37_2659_1914}

2. \begin{figure}[h] \end{figure} A smooth bead of weight 12 N is threaded onto a light elastic string of natural length 3 m . The points \(A\) and \(B\) are on a horizontal ceiling, with \(A B = 3 \mathrm {~m}\). One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). The bead hangs freely in equilibrium, 2 m below the ceiling, as shown in Figure 2.

1. Find the tension in the string.
2. Show that the modulus of elasticity of the string is 11.25 N . The bead is now pulled down to a point vertically below its equilibrium position and released from rest.
3. Find the elastic energy stored in the string at the instant when the bead is moving at its maximum speed.

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 \(C\).

4. (a) Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius \(a\) is a distance \(\frac { 3 } { 8 } a\) from the centre of its plane face.
[0pt] [You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ] \begin{figure}[h] \end{figure} A uniform solid hemisphere has mass \(m\) and radius \(a\). A particle of mass \(k m\) is attached to a point \(A\) on the circumference of the plane face of the hemisphere to form the loaded solid \(S\). The centre of the plane face of the hemisphere is the point \(O\), as shown in Figure 4. The loaded solid \(S\) is placed on a horizontal plane. The curved surface of \(S\) is in contact with the plane and \(S\) rests in equilibrium with \(O A\) making an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \sqrt { 3 }\)
(b) Find the exact value of \(k\).
\includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-09_2255_50_314_34}

5. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis in the positive \(x\) direction. At time \(t\) seconds, \(t \geqslant 1 , P\) is \(x\) metres from the origin \(O\) and is moving with speed \(v \mathrm {~ms} ^ { - 1 }\). The resultant force acting on \(P\) has magnitude \(\frac { 2 } { x ^ { 3 } } \mathrm {~N}\) and is directed towards \(O\). When \(t = 1 , x = 1\) and \(v = 3\)
Show that

1. \(v ^ { 2 } = \frac { 4 } { x ^ { 2 } } + 5\)
2. \(t = \frac { a + \sqrt { b x ^ { 2 } + c } } { d }\), where \(a , b , c\) and \(d\) are integers to be found.
   \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-13_2255_50_314_34}
   

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6. A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 3 } { 4 } \mathrm { mg }\). A particle \(P\) of mass \(m\) is attached to one end of the string. The other end of the string is attached to a fixed point \(A\). Particle \(P\) hangs freely in equilibrium at the point \(O\), vertically below \(A\).

1. Find the distance \(O A\). The particle \(P\) is now pulled vertically down to a point \(B\), where \(A B = 3 a\), and released from rest.
2. Show that, throughout the subsequent motion, \(P\) performs only simple harmonic motion, justifying your answer. The point \(C\) is vertically below \(A\), where \(A C = 2 a\).
   Find, in terms of \(a\) and \(g\),
3. the speed of \(P\) at the instant that it passes through \(C\),
4. the time taken for \(P\) to move directly from \(B\) to \(C\). \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-17_2255_50_314_34}
   

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7. \begin{figure}[h] \end{figure} A particle of mass \(m\) is attached to one end of a light inextensible string of length 8a. The other end of the string is fixed to the point \(O\) on the smooth horizontal surface of a desk. The point \(E\) is on the edge of the desk, where \(O E = 5 a\) and \(O E\) is perpendicular to the edge of the desk. The particle is held at the point \(A\), vertically above \(O\), with the string taut. The particle is projected horizontally from \(A\) with speed \(\sqrt { 8 a g }\) in the direction \(O E\), as shown in Figure 5. When the particle is above the level of \(O E\) the particle is moving in a vertical circle with radius \(8 a\). Given that, when the string makes an angle \(\theta\) with the upward vertical through \(O\), the tension in the string is \(T\),

1. show that \(T = 3 m g ( 1 - \cos \theta )\) At the instant when the string is horizontal, the particle passes through the point \(B\).
2. Find the instantaneous change in the tension in the string as the particle passes through \(B\). The particle hits the vertical side \(E F\) of the desk and rebounds. As a result of the impact, the particle loses one third of the kinetic energy it had immediately before the impact. In the subsequent motion the string becomes slack when it makes an angle \(\alpha\) with the upward vertical through \(O\).
3. Show that \(\cos \alpha = \frac { 7 } { 12 }\) DO NOT WRITEIN THIS AREA
   

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   \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-23_2255_50_314_34}
   

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   Q7

   
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