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t end. The greit rce on the d s center of as Pot OLUTION Conceptualize Imagine what happens to the rod in the figure when Ris released. rotates around the pivot at the let end. Categorize The rod is categorized asa ripid object under a toroue. The toroue is due only to the gravitational force on the rod if the rotation axis is chosen to pass through the pivot in the figure. We cannot categoriae the rod as a rigid object under constant angular acceleration because the torque exerted on the red and therefore the angular acceleration of the rod vary with its angular position analyze The only force contributing to the torgue about an axis through the pivot is the gravitational force M exerted on the rod. (The force exerted by the pivot on the red has zero torgue about the pivot because its mement arm is cEVJ To compute the torque on the rod, we assume the gravitational force acts at the center of mass of the rod as shown in the figure. Use the following as necessary: M,9. 4, and a) wite an expression for the magnitude of the net external torgue due to the gravitational force about an axis through the pivot se the equation leto obtain the anguler acceleration of the rod (in rads, using the moment of inertia for the rod, se the equation a, withLte ind the intial translational acceleration (in ms)of the right end of the rod Inalize These values are the intial values of the angular and translational accelerations. Once the rod begins te retate, the gravitational force is no longer perpendicular to the rod and the values of the twe accelerations . going to aere at the moment the rod passes through the vertical orientation. XERCISE natead of starting ata horizontal position, what if the rod started at an angle 40 from the vertical and is released from rest What will be the angular speed (in rad/) of the rod at the lowest point given that -0.2 m Hint

t end. The greit rce on the d s center of as Pot OLUTION Conceptualize Imagine what happens to the rod in the figure when Ris released. rotates around the pivot at the let end. Categorize The rod is categorized asa ripid object under a toroue. The toroue is due only to the gravitational force on the rod if the rotation axis is chosen to pass through the pivot in the figure. We cannot categoriae the rod as a rigid object under constant angular acceleration because the torque exerted on the red and therefore the angular acceleration of the rod vary with its angular position analyze The only force contributing to the torgue about an axis through the pivot is the gravitational force M exerted on the rod. (The force exerted by the pivot on the red has zero torgue about the pivot because its mement arm is cEVJ To compute the torque on the rod, we assume the gravitational force acts at the center of mass of the rod as shown in the figure. Use the following as necessary: M,9. 4, and a) wite an expression for the magnitude of the net external torgue due to the gravitational force about an axis through the pivot se the equation leto obtain the anguler acceleration of the rod (in rads, using the moment of inertia for the rod, se the equation a, withLte ind the intial translational acceleration (in ms)of the right end of the rod Inalize These values are the intial values of the angular and translational accelerations. Once the rod begins te retate, the gravitational force is no longer perpendicular to the rod and the values of the twe accelerations . going to aere at the moment the rod passes through the vertical orientation. XERCISE natead of starting ata horizontal position, what if the rod started at an angle 40 from the vertical and is released from rest What will be the angular speed (in rad/) of the rod at the lowest point given that -0.2 m Hint

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Rotating Rod A uniform rod of length L = 2.5 m and mass M = 2.8 kg is attached at one end to a frictionless pivot and is free to rotate about the pivot in the vertical plane as in the figure. The rod is released from rest in the horizontal position. What are the initial angular acceleration of the rod and the initial translational acceleration of its right end? A rod is free to rotate around a pivot at the left end. The gravitational force on the rod acts at its center of mass. Pivot SOLUTION Conceptualize Imagine what happens to the rod in the figure when it is released. It rotates --Select- around the pivot at the left end. Categorize The rod is categorized as a rigid object under a --Select-- 7torque. The torque is due only to the gravitational force on the rod if the rotation axis is chosen to pass through the pivot in the figure. We cannot categorize the rod as a rigid object under constant angular acceleration because the torque exerted on the rod and therefore the angular acceleration of the rod vary with its angular position. Analyze The only force contributing to the torque about an axis through the pivot is the gravitational force Mg exerted on the rod. (The force exerted by the pivot on the rod has zero torque about the pivot because its moment arm is (-Select- 1) To compute the torque on the rod, we assume the gravitational force acts at the center of mass of the rod as shown in the figure. (Use the following as necessary: M, g, L, and a.) Write an expression for the magnitude of the net external torque due to the gravitational force about an axis through the pivot: Use the equation = Ia to obtain the angular acceleration of the rod (in rad/s), using the moment of inertia for the rod, I ext Mg= Text = rad/s Use the equation a, = ra with r= L to find the initial translational acceleration (in m/s) of the right end of the rod: 1m/s? a.- La = Finalize These values are the initial values of the angular and translational accelerations. Once the rod begins to rotate, the gravitational force is no longer perpendicular to the rod and the values of the two accelerations -Select--- going to zero at the moment the rod passes through the vertical orientation. EXERCISE Instead of starting at a horizontal position, what if the rod started at an angle 40° from the vertical and is released from rest? What will be the angular speed (in rad/s) of the rod at the lowest point given that L = 0.82 m? Hint rad/s
Transcribed Image Text:Rotating Rod A uniform rod of length L = 2.5 m and mass M = 2.8 kg is attached at one end to a frictionless pivot and is free to rotate about the pivot in the vertical plane as in the figure. The rod is released from rest in the horizontal position. What are the initial angular acceleration of the rod and the initial translational acceleration of its right end? A rod is free to rotate around a pivot at the left end. The gravitational force on the rod acts at its center of mass. Pivot SOLUTION Conceptualize Imagine what happens to the rod in the figure when it is released. It rotates --Select- around the pivot at the left end. Categorize The rod is categorized as a rigid object under a --Select-- 7torque. The torque is due only to the gravitational force on the rod if the rotation axis is chosen to pass through the pivot in the figure. We cannot categorize the rod as a rigid object under constant angular acceleration because the torque exerted on the rod and therefore the angular acceleration of the rod vary with its angular position. Analyze The only force contributing to the torque about an axis through the pivot is the gravitational force Mg exerted on the rod. (The force exerted by the pivot on the rod has zero torque about the pivot because its moment arm is (-Select- 1) To compute the torque on the rod, we assume the gravitational force acts at the center of mass of the rod as shown in the figure. (Use the following as necessary: M, g, L, and a.) Write an expression for the magnitude of the net external torque due to the gravitational force about an axis through the pivot: Use the equation = Ia to obtain the angular acceleration of the rod (in rad/s), using the moment of inertia for the rod, I ext Mg= Text = rad/s Use the equation a, = ra with r= L to find the initial translational acceleration (in m/s) of the right end of the rod: 1m/s? a.- La = Finalize These values are the initial values of the angular and translational accelerations. Once the rod begins to rotate, the gravitational force is no longer perpendicular to the rod and the values of the two accelerations -Select--- going to zero at the moment the rod passes through the vertical orientation. EXERCISE Instead of starting at a horizontal position, what if the rod started at an angle 40° from the vertical and is released from rest? What will be the angular speed (in rad/s) of the rod at the lowest point given that L = 0.82 m? Hint rad/s
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