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11.6: Chapter 11 Homework Problems

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    Exercise \(\PageIndex{1}\)

    You are designing a bench grinder with an operating speed of 3600 rpm.

    • If you want the grinder to reach its full operating speed in 4 seconds, what must the rate of angular acceleration be in radians per second squared?
    • If the grinding wheel has a diameter of 8 inches, what will the speed of the surface of the wheel be?
    A bench grinder, consisting of a motor attached to two grinding wheels.
    Figure \(\PageIndex{1}\): A bench grinder.
    Solution

    \(\alpha = 94.25 \ \frac{rad}{s}\)

    \(v = 125.67 \ ft/s\)

    Exercise \(\PageIndex{2}\)

    A belt-driven system has an input at pulley A, which drives pulley B, which is attached with a solid shaft to pulley C, which drives pulley D. If the input is rotating at 60 rad/s counterclockwise, determine the angular velocity and direction of rotation for the output at D.

    Pulley A, with a diameter of 5 inches, is connected by a belt loop to pulley B, which has a diameter of 2 inches. Pulley B is mounted on the same shaft as pulley C, which has a diameter of 6 inches. A belt loop connects pulley C to pulley D, which has a diameter of 3 inches. The input of this system consists of pulley A rotating counterclockwise at 60 radians/second.
    Figure \(\PageIndex{2}\): problem diagram for Exercise \(\PageIndex{2}\). A four-pulley system in which A and B are attached by a belt, C and D are attached by another belt, and B and C are mounted on the same shaft.
    Solution

    \(\omega_D = 300 \ \frac{rad}{s}\) counterclockwise

    Exercise \(\PageIndex{3}\)

    The piston in a piston and crank mechanism has the velocity and acceleration shown below. Using absolute motion analysis, determine the current angular velocity and angular acceleration for the crank.

    Side view of a piston (represented as a rectangle at the top of the diagram) and a crank (represented as a 150-mm-radius circle at the bottom), connected by a bar that is rigidly attached to the piston's side at one end and to a point on the outer edge of the crank at the other. The piston is descending at a velocity of 2 m/s and an acceleration of 5 m/s², with its motion causing the crank to rotate clockwise due to the bar. Currently, the crank's rotation is such that its point where the bar attaches is on the rightmost edge of the circle, and there is a vertical distance of 400 mm between the bar attachment point on the piston and the center of the crank.
    Figure \(\PageIndex{3}\): problem diagram for Exercise \(\PageIndex{3}\). A piston descends, causing the crank mounted on a fixed axle below it to rotate due to the bar that connects the piston and the crank.
    Solution

    \(\omega = 13.33 \ \frac{rad}{s}\) clockwise

    \(\alpha = 100.16 \ \frac{rad}{s^2}\) clockwise

    Exercise \(\PageIndex{4}\)

    A trapdoor is being opened with a hydraulic cylinder extending at constant rate of 0.7 m/s. If the door is currently at a twenty-degree angle as shown below, what is the current angular velocity and angular acceleration for the door?

    Side view of a trapdoor, depicted as a 2-meter-long bar attached to the ground via hinge at its left side. One end of a hydraulic cylinder is attached to the midpoint of the bar. The other end of the cylinder is attached to a fixed point on the ground, 1.5 meters to the right of the trapdoor's hinge. The cylinder extends to open the door; currently the door is partially opened so it makes a 20° angle with the ground.
    Figure \(\PageIndex{4}\): problem diagram for Exercise \(\PageIndex{4}\). Side view of a trapdoor being raised by the extension of a hydraulic cylinder that has one end attached to the door and the other fixed to a point on the ground.
    Solution

    \(\dot{\theta} = 0.896 \ \frac{rad}{s}, \, \ddot{\theta} = -1.246 \ \frac{rad}{s^2} \)

    Exercise \(\PageIndex{5}\)

    A robotic arm experiences the angular velocities and accelerations shown below. Based on this information, determine the velocity and the acceleration of the end of the arm in the \(x\) and \(y\) directions.

    A two-member robotic arm: the first member, 3 feet long, extends upwards and to the right from the left endpoint, which is attached to a fixed base and forms the origin of a standard-orientation Cartesian coordinate system. A motor at the joint between the member and the base provides a clockwise rotation at a steady rate of 2 rad/s. This first member makes an angle of theta = 65° above the x-axis. The second member, 2 feet long, extends down and to the right from the right end of the 3-foot member, making an angle of phi = 30° below the horizontal. A motor at the joint between the two members provides a counterclockwise rotation with a velocity of 4 rad/s and an acceleration of -1 rad/s².
    Figure \(\PageIndex{5}\): problem diagram for Exercise \(\PageIndex{5}\). A two-member robotic arm attached to a fixed base experiences rotation from two motors located at the joints.
    Solution

    \( v_x = 9.44 \ ft/s, \, v_y = 4.39 \ ft/s \)

    \( a_x = -33.78 \ ft/s^2, \, a_y = 3.39 \ ft/s^2 \)

    This page titled 11.6: Chapter 11 Homework Problems is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jacob Moore & Contributors (Mechanics Map) via source content that was edited to the style and standards of the LibreTexts platform.