OCR M3 (Mechanics 3) 2008 June

1 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 1.8 m and modulus of elasticity 1.35 mg N . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. \(P\) is held at rest at a point on the surface 3 m from \(O\). The particle is then released. Find

1. the initial acceleration of \(P\),
2. the speed of \(P\) at the instant the string becomes slack.

2 A particle \(P\) of mass 0.2 kg is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits a horizontal smooth surface. The direction of motion of \(P\) immediately before impact makes an angle of \(27 ^ { \circ }\) with the surface. Given that the coefficient of restitution between the particle and the surface is 0.6 , find

1. the vertical component of the velocity of \(P\) immediately after impact,
2. the magnitude of the impulse exerted on \(P\).

3
\includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-2_387_561_1055_794} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 0.8 kg and 2.0 kg respectively. The spheres are on a horizontal surface. \(A\) is moving with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the line of centres when it collides with \(B\), which is stationary (see diagram). The coefficient of restitution between the spheres is 0.75 . Find the speed and direction of motion of \(A\) immediately after the collision.

4 A particle \(P\) of mass \(m \mathrm {~kg}\) is held at rest at a point \(O\) on a fixed plane inclined at an angle \(\sin ^ { - 1 } \left( \frac { 4 } { 7 } \right)\) to the horizontal. \(P\) is released and moves down the plane. The total resistance acting on \(P\) is \(0.2 m v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\).

1. Show that \(5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = 28 - v\) and hence find an expression for \(v\) in terms of \(t\).
2. Find the acceleration of \(P\) when \(t = 10\).

5
\includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-3_581_903_267_621} Two uniform rods \(X A\) and \(X B\) are freely jointed at \(X\). The lengths of the rods are 1.5 m and 1.3 m respectively, and their weights are 150 N and 130 N respectively. The rods are in equilibrium in a vertical plane with \(A\) and \(B\) in contact with a rough horizontal surface. \(A\) and \(B\) are at distances horizontally from \(X\) of 0.9 m and 0.5 m respectively, and \(X\) is 1.2 m above the surface (see diagram).

1. The normal components of the contact forces acting on the rods at \(A\) and \(B\) are \(R _ { A } \mathrm {~N}\) and \(R _ { B } \mathrm {~N}\) respectively. Show that \(R _ { A } = 125\) and find \(R _ { B }\).
2. Find the frictional components of the contact forces acting on the rods at \(A\) and \(B\).
3. Find the horizontal and vertical components of the force exerted on \(X A\) at \(X\), stating their directions.

6 A particle \(P\) of mass 0.1 kg moves in a straight line on a smooth horizontal surface. A force of \(( 0.36 - 0.144 x ) \mathrm { N }\) acts on \(P\) in the direction from \(O\) to \(P\), where \(x \mathrm {~m}\) is the displacement of \(P\) from a point \(O\) on the surface at time \(t \mathrm {~s}\).

1. By using the substitution \(x = y + 2.5\), or otherwise, show that \(P\) moves with simple harmonic motion of period 5.24 s , correct to 3 significant figures. The maximum value of \(x\) during the motion is 3 .
2. Write down the amplitude of \(P\) 's motion and find the two possible values of \(x\) for which \(P\) 's speed is \(0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. On each of the first two occasions when \(P\) has speed \(0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 } , P\) is moving towards \(O\). Find the time interval between
   (a) these first two occasions,
   (b) the second and third occasions when \(P\) has speed \(0.48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

7
\includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-4_517_677_267_733} A particle \(P\) of mass \(m \mathrm {~kg}\) is slightly disturbed from rest at the highest point on the surface of a smooth fixed sphere of radius \(a\) m and centre \(O\). The particle starts to move downwards on the surface. While \(P\) remains on the surface \(O P\) makes an angle of \(\theta\) radians with the upward vertical and has angular speed \(\omega\) rad s \(^ { - 1 }\) (see diagram). The sphere exerts a force of magnitude \(R \mathrm {~N}\) on \(P\).

1. Show that \(a \omega ^ { 2 } = 2 g ( 1 - \cos \theta )\).
2. Find an expression for \(R\) in terms of \(m , g\) and \(\theta\). At the instant that \(P\) loses contact with the surface of the sphere, find
3. the transverse component of the acceleration of \(P\),
4. the rate of change of \(R\) with respect to time \(t\), in terms of \(m , g\) and \(a\).