Question

A marching band consists of rows of musicians walking in straight, even lines. When a marching band performs in an event, such as a parade, and must round a curve in the road, the musician on the outside of the curve must walk around the curve in the same amount of time as the musician on the inside of the curve. This motion can be approximated by a disk rotating at a constant rate about an axis perpendicular to its plane. In this case, the axis of rotation is at the inside of the curve.Consider two musicians, Alf and Beth. Beth is four times the distance from the inside of the curve as Alf.


Knowing that If Beth travels a distance s during time Δt, how far does Alf travel during the same amount of time= (1/4)s


If Alf moves with speed v, what is Beth's speed? Speed in this case means the magnitude of the linear velocity, not the magnitude of the angular velocity.


a)4v b) v c) v/4

Answer 1

Answer:

$S_A=S/4$

$V_B=4*V_A$

Explanation:

We know that $R_B=4*R_A$ and $\omega_A=\omega_B=\omega$

The distance traveled by Beth is given by:

$S_B=S=V_B*\Delta t=\omega*R_B*\Delta t$

Replacing the relation between the radius of both paths:

$S=4*\omega*R_A*\Delta t$                          (eq1)

The distance traveled by Alf is given by:

$S_A=V_A*\Delta t=\omega*R_A*\Delta t$     (eq2)

If we replace eq2 into eq1:

$S=4*S_A$   therefore: $S_A=S/4$

The relation between speeds is:

$\omega_A=\omega_B$

$\frac{V_A}{R_A} =\frac{V_B}{R_B}$

$\frac{V_A}{R_A} =\frac{V_B}{4*R_A}$

$V_A =\frac{V_B}{4}$

$V_B =4*V_A$