12.5: Chapter 12 Homework Problems

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Exercise \(\PageIndex{1}\)

The SUV shown below has an initial velocity of 90 ft/s. It slams on its brakes, coming to a stop over a distance of 300 feet. If the car has a weight of 3500 lbs and a center of mass as shown below, what are the normal forces at the front wheels? What are the normal forces at the back wheels?

Side view of an SUV facing right, traveling towards the right at an initial velocity of 90 ft/s. There is a distance of 8 feet between its front and rear wheels. The center of mass is located at the midpoint between the wheels, at a distance of 2.5 feet above the ground. — Figure \(\PageIndex{1}\): problem diagram for Exercise \(\PageIndex{1}\). A car traveling in a straight line applies its brakes, coming to a gradual stop.

Solution

\( F_{N_{rear}} = 1291.4 \ lbs\)

\( F_{N_{front}} = 2208.6 \ lbs\)

Exercise \(\PageIndex{2}\)

A ring-shaped space station can be approximated as a thin ring 60 meters in diameter with a mass of 500,000 kg. The space station has a set of thrusters able to exert equal and opposite forces as shown below. If we want to cause an angular acceleration of 0.1 rad/s² in the space station, what is the force required from each thruster?

A circular ring, representing the space station, is rotating counterclockwise with an angular acceleration of 0.1 rad/s². Two thrusters located on the outside of the ring, at its leftmost and rightmost points, exert the same magnitude of force in opposite directions. The thruster on the right exerts a force that points towards the top of the page, and the thruster on the left exerts a force that points towards the bottom of the page. — Figure \(\PageIndex{2}\): problem diagram for Exercise \(\PageIndex{2}\). A ring-shaped space station is given the specified angular acceleration through the firing of two thrusters located on either side of a diameter, pointing in opposite directions.

Solution: \(F_{thruster} = 750 \ kN\)

Exercise \(\PageIndex{3}\)

A 50-kg barrel with a diameter of 0.75 meters is placed on a 20-degree slope. Assuming the barrel rolls without slipping, what will the acceleration of the barrel's center of mass be?

Side view of a 50-kg barrel at the top of a 20° incline. — Figure \(\PageIndex{3}\): problem diagram for Exercise \(\PageIndex{3}\). A 50-kg barrel with a 0.75-meter diameter rolls down a 20° incline without slipping.

Solution: \(a_x = 2.24 \ m/s^2\)

Exercise \(\PageIndex{4}\)

A 3-meter-long, 25-kg beam is supported by two cables as shown below. You can treat the beam as a slender rod. Assume that we want the left end of the beam at point A to remain at a constant height while the right end of the beam at point B accelerates upwards at a rate of 1 m/s².

What is the rate of acceleration of the center of the beam and the rate of angular acceleration for the beam?
What will \(T_1\) and \(T_2\) need to be to achieve these accelerations?

A horizontal beam with left endpoint A, midpoint C, and right endpoint B is held in the air by two vertical cables. The first cable is attached to the point on the beam 0.5 meters to the right of A, with the upwards tension force on the beam from the cable labeled as T1. The second cable is attached to the point on the beam 1 meter to the left of B, with the upwards tension force on the beam from the cable labeled as T2. — Figure \(\PageIndex{4}\): problem diagram for Exercise \(\PageIndex{4}\). A horizontal beam is held in the air by the tension forces from two vertical cables attached near the beam's ends.

Solution

\( a_{C\y} = 0.5 \ m/s^2, \, \alpha = 0.333 \ \frac{rad}{s} \)

\( T_1 = 81.75 \ N, \, T_2 = 176 \ N \)

Exercise \(\PageIndex{5}\)

You are modeling the robotic arm shown below. Treat each section of the arm as a slender rod. Section OA weighs 30 lbs and section AB weighs 18 lbs. If we want the relative angular accelerations and velocities shown below, what should the motor torques be at O and A? (This is a top-down view of the robot arm.)

A two-segment robotic arm: the first segment is horizontal and 3 feet long, with its left endpoint O attached to a fixed base and its right endpoint A being the point of attachment for the second segment AB. The second segment is 2 feet long, extending downward and to the right at an angle of 30° below the horizontal. Point O experiences a counterclockwise rotation, at a relative angular velocity of 5 rad/s and an angular acceleration of 1 rad/s². Point A experiences a counterclockwise rotation, at a relative angular velocity of 3 rad/s and angular acceleration of 0 rad/s². — Figure \(\PageIndex{5}\): problem diagram for Exercise \(\PageIndex{5}\). Top-down view of a two-segment robotic arm with one end attached to a fixed base, with motors at the two joints providing rotation.

Solution

\(M_O = - 3.9 \ ft\)-\(lbs\)

\(M_A = -19.3 \ ft\)-\(lbs\)