A*R*C*S 12.7.1. Friction on a Baseball.
Find a symbolic expression for the frictional force exerted by a baseball pitcher (tangent to the surface of the baseball) when throwing a fastball that spins at \(2500 \mathrm{~rpm}\text{.}\)
Section 12.7 Practice - Torque
Subsection A*R*C*S Practice
A*R*C*S 12.7.2. String Around a Tennis Ball.
A string is wrapped many times around a tennis ball that has mass \(m = 60 \mathrm{~g}\) and radius \(r = 7 \mathrm{~cm}\text{.}\) For a hollow ball, \(I = \frac{2}{3}mr^2\) about the center or \(I = \frac{5}{3}mr^2\) about the edge. The string is attached to the ceiling and the ball is allowed to fall (see the figure below). Determine the tension in the string.
Solution.
There are only two forces acting on the ball: the gravitational force with magnitude \(mg\) and the tension force with unknown magnitude \(F^T\text{.}\) The ball accelerates downward, so the net force should also point downward, leading to the conclusion that the gravitational force is larger than the tension force.
Using the Rotational Law of Motion for an axis passing through the place where the string is in contact with the ball:
\begin{equation*}
\vec{\tau}_{net} = I\vec{\alpha}
\end{equation*}
\begin{equation*}
mgr = \frac{5}{3}mr^2\frac{a}{r}
\end{equation*}
\begin{equation*}
a = \frac{3}{5}g
\end{equation*}
Now using the regular Law of Motion (with negative, downward acceleration):
\begin{equation*}
F^T - mg = -ma
\end{equation*}
\begin{equation*}
F^T = \frac{2}{5}mg
\end{equation*}
Plugging in the given numbers gives:
\begin{equation*}
F^T = 0.24 \mathrm{~N}
\end{equation*}
Subsection Explanation Practice
Explanation 12.7.3. String Around a Tennis Ball II.
Suppose you were to drop another tennis ball with the same mass and radius as in the previous activity, but a smaller moment of inertia (this tennis ball might be solid instead of hollow). Is the magnitude of the translational acceleration of the new tennis ball greater than, less than, or equal to the magnitude of the translational acceleration of the original tennis ball?
Explanation 12.7.4. Tipping Truck.
You observe a truck driving along a road at a constant speed \(v\text{.}\) As the truck begins to turn along a flat, circular part of the road, it also begins to tip over. Use an extended free body diagram to explain why the truck begins to tip.
Subsection Numerical Practice
Calculation 12.7.5. Torque.
A force is applied on a lever arm \(1 \mathrm{~m}\) away from the pivot point, and it produces torque. How much force would have to be applied to produce the same amount of torque if it were applied \(4 \mathrm{~m}\) from the pivot point? Assume that both forces are applied perpendicularly to the lever arm.
- Four times the initial force
- Sixteen times the initial force
- One fourth of the initial force
- One sixteenth of the initial force
- Same as the initial force
Answer.
C.
Calculation 12.7.6. See-Saw.
Two children of different weights are riding a seesaw. How do they position themselves with respect to the pivot point (the fulcrum) so that they are balanced?
- The heavier child sits closer to the fulcrum.
- The heavier child sits farther from the fulcrum.
- Both children sit at equal distance from the fulcrum.
- Since both have different weights, they will never be in balance.
Answer.
A.
Calculation 12.7.7. Angular Acceleration.
A uniform, solid disk with a mass of \(24.3 \mathrm{~kg}\) and a radius of \(0.314 \mathrm{~m}\) is oriented vertically and is free to rotate about a frictionless axle. Forces of \(90 \mathrm{~N}\) and \(125 \mathrm{~N}\) are applied to the disk in the same horizontal direction, but one force is applied to the top and the other is applied to the bottom. What is the magnitude of the angular acceleration of the disk?
Answer.
\(9.17 \mathrm{~rad/s^2}\)
References References
[1]
Numerical practice activities provided by BoxSand: https://boxsand.physics.oregonstate.edu/welcome.
