Solve the problem.
-A wheel with radius $2 \mathrm {~m}$ rolls at $12 \mathrm { rad } / \mathrm { s }$. How fast is a point on the rim of the wheel rising when the point is $\pi / 3$ radians above the horizontal (and rising)? (Round your answer to one decimal place.)
A) $48.0 \mathrm {~m} / \mathrm { s }$
B) $12.0 \mathrm {~m} / \mathrm { s }$
C) $24.0 \mathrm {~m} / \mathrm { s }$
D) $6.0 \mathrm {~m} / \mathrm { s }$

Question

Quizplus · Accepted Answer

Let the point on the rim of the wheel be P, and let O be the center of the wheel. Let the horizontal line through O be the x-axis and the vertical line through O be the y-axis. Let $\theta$ be the angle between the radius that passes through P and the positive x-axis, as shown in the diagram below.

[asy]
size(150);
draw((0,-3)--(0,3),black+linewidth(1.5),EndArrow(size=5));
draw((-3,0)--(3,0),black+linewidth(1.5),EndArrow(size=5));
draw(shift((0,0))*rotate(60)*yscale(2)*arc((0,0),1,0,180),black+linewidth(1.5));
dot((0,0));
label("$O$",(0,0),SW);
pair P = (-1.1547,1);
draw(O--P,black+linewidth(1.5));
dot(P);
label("$P$",P,NW);
draw(P--(P.x,0),dashed);
label("$\theta$",(0.5,0.3));
draw(rightanglemark(O,P,(P.x,0),3.5));
[/asy]

Since the wheel is rolling along the ground without slipping, the distance that a point on the rim travels is the same as the distance that the center of the wheel travels. Therefore, the distance that P travels in one second is the same as the magnitude of the velocity of O, which is $r\omega = 2\times 12 = 24$ m/s.

To find how fast P is rising, we want to find the derivative of the height of P with respect to time. From the diagram, we see that the height of P is given by $h = 2\sin\theta$. Therefore, the derivative of the height of P with respect to time is 
$$\frac{dh}{dt} = \frac{dh}{d\theta} \cdot \frac{d\theta}{dt} = 2\cos\theta \cdot \frac{d\theta}{dt}.$$
When $\theta = \pi/3$, we have $\cos\theta = 1/2$, so
$$\frac{dh}{dt} = 2\cdot \frac{1}{2} \cdot 12 = \boxed{12}\text{ m/s}.$$

Solve the Problem 2 m2 \mathrm {~m}2 m Rolls At 12rad/s12 \mathrm { rad } / \mathrm { s }12rad/s

Solve the Problem $2 \mathrm {~m}$ Rolls At $12 \mathrm { rad } / \mathrm { s }$