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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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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, exerts a force T, to the right on the cylinder. A rope wrapped around the core, which has radius R, exerts a force T downward on the cylinder. A solid cylinder pivoted about the z-axis through O. The moment arm of T, is R, and the moment arm of T, is Ry R Ra (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, could be applied by a drive belt wrapped around the drum. The force T, 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: r = rF sin(0) = Fd. (Use the following as necessary: T, T,, R,, and R,) The torque due to T, about the rotation axis is 7, = - (The sign is negative because the torque tends to produce clockwise rotation.) The torque due to T, is , = + . (The sign is positive because the torque tends to produce counterclockwise rotation of the cylinder.) Evaluate the net torque about the rotation axis: As a quick check, notice that if the two forces are of equal magnitude, the net torque is negative because R, V R.. Starting from rest with both forces of equal magnitude acting on it, the cylinder would rotate clockwise because T, would be more effective at turning it than would T,. (b) Suppose T, = 4.3 N, R, = 1.30 m, T, = 6.0 N, and R, = 1.10 m. What is the net torque about the rotation axis (in N- m), and which way does the cylinder rotate starting from rest? SOLUTION Substitute the given values: Σ N. m The net torque above indicates the cylinder rotates --Select-- from rest. EXERCISE What if the forces T, and T, both reverse direction, and with new magnitudes T, = 5.7 N and T, = 12.4 N. What is the magnitude of the net torque now (in N. m)? (Use the same sign convention for positive and negative torques. The radii R, and R, are 1.30 m and 1.10 m as given in the example part (b).) Hint N.m net
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, exerts a force T, to the right on the cylinder. A rope wrapped around the core, which has radius R, exerts a force T downward on the cylinder. A solid cylinder pivoted about the z-axis through O. The moment arm of T, is R, and the moment arm of T, is Ry R Ra (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, could be applied by a drive belt wrapped around the drum. The force T, 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: r = rF sin(0) = Fd. (Use the following as necessary: T, T,, R,, and R,) The torque due to T, about the rotation axis is 7, = - (The sign is negative because the torque tends to produce clockwise rotation.) The torque due to T, is , = + . (The sign is positive because the torque tends to produce counterclockwise rotation of the cylinder.) Evaluate the net torque about the rotation axis: As a quick check, notice that if the two forces are of equal magnitude, the net torque is negative because R, V R.. Starting from rest with both forces of equal magnitude acting on it, the cylinder would rotate clockwise because T, would be more effective at turning it than would T,. (b) Suppose T, = 4.3 N, R, = 1.30 m, T, = 6.0 N, and R, = 1.10 m. What is the net torque about the rotation axis (in N- m), and which way does the cylinder rotate starting from rest? SOLUTION Substitute the given values: Σ N. m The net torque above indicates the cylinder rotates --Select-- from rest. EXERCISE What if the forces T, and T, both reverse direction, and with new magnitudes T, = 5.7 N and T, = 12.4 N. What is the magnitude of the net torque now (in N. m)? (Use the same sign convention for positive and negative torques. The radii R, and R, are 1.30 m and 1.10 m as given in the example part (b).) Hint N.m net
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