Answer to Question #108631 in Mechanics | Relativity for Shavonne Fraley

Question

On an interstate highway, you are traveling at 30 m/sec. The diameter of the wheels is 50 cm. How fast are the wheels rotating in revolutions/sec?
c. On your vacation, you visited several casinos in Las Vegas, and in one of them, you stopped by a roulette table. You observed that roulette wheel takes about 25 seconds to come to a stop, and it completed 40 revolutions in this time. How fast was it rotating at the very beginning before starting to slow down?
2.
a. Provide a mathematical definition of momentum, and provide its unit in the
MKS system of units.
b. In a game of billiards, there are usually 11 identical balls. In this problem,
we concentrate on two of them. The first one is hit by the player and moves at a speed of 14 m/sec, and it collides with a second stationary ball. The two balls stick together. What is the speed of the balls after the collision?

Answer 1

On an interstate highway, you are traveling at 30 m/sec. The diameter of the wheels is 50 cm. How fast are the wheels rotating in revolutions/sec?

From,

$\omega=\frac{v}{r}$

where v is the velocity r is the radius and $\omega$ is the angular velocity

$\omega=\frac{v}{r}=\frac{30}{0.25}=120rads^{-1}=>\frac{120}{2\pi}=19.09rev/sec$

c. On your vacation, you visited several casinos in Las Vegas, and in one of them, you stopped by a roulette table. You observed that roulette wheel takes about 25 seconds to come to a stop, and it completed 40 revolutions in this time. How fast was it rotating at the very beginning before starting to slow down?

$\theta=\omega_0\times t+\frac{1}{2}\alpha t^2 \to(1)$

$\omega=\omega_0+{}\alpha t\to(2)$

where ,$\theta$ is the angle it rotated ,$\omega_0$ is the intial angular velocity and $\alpha$ is the angular acceleration

From 2,

$0=\omega_0+\alpha (25) => \alpha=-\frac{\omega_0}{25}$

substituting to (1)

$40\times2\pi=\omega_0\times25+0.5\times (-\frac{\omega_0}{25})\times25^2$

solving this gives,

$\omega_0=20.10rad/s$

a. Provide a mathematical definition of momentum, and provide its unit in the

MKS system of units.

Momentum is a vector quantity that is the product of the mass and the velocity of an object or particle.

$Momentum (P) = Mass(m) \times Velocity (V)$

units in MKS = $kg\ ms^{-1}$

b. In a game of billiards, there are usually 11 identical balls. In this problem,

we concentrate on two of them. The first one is hit by the player and moves at a speed of 14 m/sec, and it collides with a second stationary ball. The two balls stick together. What is the speed of the balls after the collision?

since balls are identical the mass of the balls are equal,

$m_1=m_2=m$

From the momentum conservation principle, let $V_1$ be the first ball speed and $V_2$ be the speed of the two balls after collision.

$m_1 V_1=(m_1+m_2) V_2 \\ mV_1=(2m)V_2\\ V_2=\frac{V_1}{2}=\frac{14}{2}=7\ m/sec$