Question

1.A cyclist negotiates a curve of radius 100m with a velocity 10 m/s. How much should he

lean from the vertical in doing so?

2.. A disc revolves in a horizontal plane at a steady rate of 3 revs/s. A coin will remain on the

disc if it is kept a distance of 2 cm from the axis of rotation. What is the coefficient of

friction between the coin and the disc?

Answer 1

1) Let's apply Newton's Second Law of Motion in projections on axis $x$- and $y$:

$$Ncos\theta=mg,$$ $$Nsin\theta=\dfrac{mv^2}{r}.$$

Dividing the second equation by the first one, we get:

$$tan\theta=\dfrac{v^2}{rg},$$ $$\theta=tan^{-1}(\dfrac{v^2}{rg}),$$ $$\theta=tan^{-1}(\dfrac{(10\ \dfrac{m}{s})^2}{100\ m\cdot9.8\ \dfrac{m}{s^2}})=5.8^{\circ}.$$

2)

$$F_c=F_{fr},$$ $$\dfrac{mv^2}{r}=\mu mg,$$ $$\omega^2r=\mu g,$$ $$\mu=\dfrac{\omega^2r}{g},$$ $$\mu=\dfrac{(3\ \dfrac{rev}{s}\cdot\dfrac{2\pi\ rad}{1\ rev})^2\cdot0.02\ m}{9.8\ \dfrac{m}{s^2}}=0.72$$