Write down an expression, in terms of M, v and R, for the angular momentum of the system about the vertical axis just before the collision.

Just after the collision the system begins to rotate about the vertical axis with angular velocity ω. Show that the angular momentum of the system is equal to \(\frac{4}{3}M{R^2}\omega \).

Hence, show that \(\omega  = \frac{v}{{4R}}\).

Determine in terms of M and v the energy lost during the collision.

Show that the angular deceleration of the system is 0.043 rad\(\,\)s^–2.

Calculate the number of revolutions made by the system before it comes to rest.

\(\frac{M}{3}vR\)

[1 mark]

evidence of use of: \(L = I\omega  = \left( {M{R^2} + \frac{M}{3}{R^2}} \right)\omega \)

[1 mark]

evidence of use of conservation of angular momentum, \(\frac{{MvR}}{3} = \frac{4}{3}M{R^2}\omega \)

«rearranging to get \(\omega  = \frac{v}{{4R}}\)»

[1 mark]

initial KE = \(\frac{{M{v^2}}}{6}\)

final KE = \(\frac{{M{v^2}}}{{24}}\)

energy loss = \(\frac{{M{v^2}}}{8}\)

[3 marks]

\(\alpha \) «= \(\frac{3}{4}\frac{\Gamma }{{M{R^2}}}\)» = \(\frac{3}{4}\frac{{0.01}}{{0.7 \times {{0.5}^2}}}\)

«to give \(\alpha \) = 0.04286 rad\(\,\)s⁻²»

Working OR answer to at least 3 SF must be shown

[1 mark]

\(\theta  = \frac{{\omega _i^2}}{{2\alpha }}\) «from \(\omega _f^2 = \omega _i^2 + 2\alpha \theta \)»

\(\theta \) «\( = \frac{{{v^2}}}{{32{R^2}\alpha }} = \frac{{{{2.1}^2}}}{{32 \times {{0.5}^2} \times 0.043}}\)» = 12.8 OR 12.9 «rad»

number of rotations «= \(\frac{{12.9}}{{2\pi }}\)» = 2.0 revolutions

[3 marks]