# PHYS*1080 Sample Quiz 7

1. A stone of mass $200\; \text{g}$ is swung on a horizontal circle of radius $1 \; \text m$ with a speed of $5\; \text{m/s}.$ (4 marks)

(a) What is the tension in the string?

(b) How much work does the tension force do in 5 revolutions?

(c) Suppose it had taken $10 \; \text s$ to get this speed from rest. What then is the angular acceleration?

Part Solution Marks
(a) $T = F = mv^2/r = 0.2 \; \text{kg} \times (5 \; \text{m/s}^2/1\; \text m = 5\; \text N)$  (1) for  $T = \frac{mv^2}{r}$
(b) Work done by tension = 0 since there is no displacement in the direction of the force.  (1)
(c) $\omega_o = 0 \quad \quad \omega = v/r = 5/1\; \text s ^{-1} \\ t = 10\; \text s \quad \; \alpha = ? \\ \alpha = (\omega-\omega_o)/t = (5-0)/10 = 0.5\; \text{rad/s}^2$

(1) $\alpha = \frac{\omega - \omega_o}{t}$

2. A figure skater rotating at a constant 1.5 revolutions per second has a rotational kinetic energy of $1500 \; \text J$. (4 marks)

(a) What is her moment of inertia?

(b) What net torque (moment) is acting on the skater?

(c) What is her angular momentum?

Part Solution Marks
(a) $K.E. = \frac{1}{2}I \omega^2 \quad \quad \therefore I = 2(K.E.)/\omega^2 \\ \therefore I = (2 \times 1500 \; \text J)/(1.5 \times 2\pi \;\text{rad/s}^2 = 33.8\; \text {kg m}^2)$ (1.0)
(0.5)
(b) $\sum \tau = l \alpha \quad \quad \alpha = 0 \\ \therefore\; \text{Net torque}\; = 0$ (1.0)
(c)

\begin {align} \text{Angular momentum} & = I\omega \\ & = 34\; \text{kg m}^2 \times 3 \pi\; \text{rad/s}\\ & = 3.2 \times 10^2 \; \text{kg m}^2 /\text s \end {align}

$(\text{or} I\omega = 2 (K.E.)/ \omega)$

(1.0)

(0.5)

3. A molecule of mass $1.0 \times 10^{-25} \; \text {kg}$ is traveling at a constant speed of $2.0 \times 10^{-6}\; \text{m/s}$ in a circle of radius $3.0 \times 10^{-8}\text m$. What is the force (magnitude and direction) on the molecule? (1.5 marks)

Solution Marks
$F = \frac{mv^2}{r} = \frac{1.0 \times 10^{-25}\; \text{kg} \times (2.0 \times 10^{-6}\; \text{m/s})^2}{3.0 \times 10^{-8}\text m}$ (0.5)
$= 1.3 \times 10^{-29}\; \text N$ (0.5)
Towards the centre of rotation (0.5)

4. The earth exerts a gravitational force $2 \times 10^{20}$ on the moon, which travels a distance $2.4 \times 10^9$ along the complete circumference of its circular orbit around the earth in  a time of $2.4 \times 10^6 \; \text s$ (about 28 days). How much work does the earth do on the moon in each orbit? (0.5 marks)