The Net Torque on a Cylinder A one-piece cylinder is shaped as shown in the figure, with a core section protruding from the larger drum. The cylinder is free to rotate about the zais shown in the drawing. A rope wrapped around the drum, which has radius R exerts a force , to the right on the cylinder. A rope wrapped around the core, which has radius R exerts a force downward on the cylinder. Asold cylinder pivoted abouthe througho. The moment am oti, and he moment am
The Net Torque on a Cylinder A one-piece cylinder is shaped as shown in the figure, with a core section protruding from the larger drum. The cylinder is free to rotate about the zais shown in the drawing. A rope wrapped around the drum, which has radius R exerts a force , to the right on the cylinder. A rope wrapped around the core, which has radius R exerts a force downward on the cylinder. Asold cylinder pivoted abouthe througho. The moment am oti, and he moment am
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
Transcribed Image Text:**The Net Torque on a Cylinder**
A one-piece cylinder is shaped as shown in the figure, with a core section protruding from the larger drum. The cylinder is free to rotate about the central z-axis shown in the drawing. A rope wrapped around the drum, which has radius \( R_1 \), exerts a force \( T_1 \) to the right on the cylinder. A rope wrapped around the core, which has radius \( R_2 \), exerts a force \( T_2 \) downward on the cylinder.
**Diagram Explanation:**
- The diagram shows a solid cylinder pivoted about the z-axis through the center.
- Labeled components include:
- \( R_1 \) and \( R_2 \) as the radii of the drum and core, respectively.
- Forces \( T_1 \) and \( T_2 \) applied via ropes.
- Points where forces are applied, with arrows indicating direction.
**(a) What is the net torque acting on the cylinder about the rotation axis (which is the z-axis in the figure)?**
**SOLUTION**
*Conceptualize*: Imagine that the cylinder in the figure is a shaft in a machine. The force \( T_1 \) could be applied by a drive belt wrapped around the drum. The force \( T_2 \) could be applied by a friction brake at the surface of the core.
*Categorize*: This example is a substitution problem in which we evaluate the net torque using the equation: \( \tau = rF \sin(\theta) = Fr \).
(Use the following as necessary: \( T_1 \), \( T_2 \), \( R_1 \), and \( R_2 \).)
The torque due to \( T_1 \) about the rotation axis is: \( \tau_1 = R_1 T_1 = \_\_\_ \). (The sign is negative because the torque tends to produce clockwise rotation.) The torque due to \( T_2 \) is: \( \tau_2 = R_2 T_2 = \_\_\_ \). (The sign is positive because the torque tends to produce counterclockwise rotation.)
Evaluate the net torque about the rotation axis:
\[
\tau_{\text{net}} = \tau_2 - \tau_1 = R_2 T_2 - R_1 T_1 =
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