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11.1.13.3: Problems

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    Exercise \(\PageIndex{1}\): Calculate the angle for a box on an inclined plane to tip

    A box of uniformly distributed mass sits in equilibrium on a ramp as shown in the animation (position is given in meters). Restart.

    1. Suppose you wish to increase the angle of the ramp. What is the maximum angle of the ramp so that the box does not tip?
    2. What is the minimum coefficient of static friction between the box and ramp in order for the box to not slide when the ramp is at the angle measured in part (a)?

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{2}\): Calculating supporting tensions in a suspended load

    A steel block sits on a board that is held by two strings. The strings are attached to a crossbar that is held by two ringstands on a laboratory table. Assume that all objects have uniformly distributed mass, which means that the center of mass of each object is at its geometric center. The masses are as follows: The block has a mass of \(2.0\text{ kg}\), the board has a mass of \(0.50\text{ kg}\), and the crossbar has a mass of \(1.0\text{ kg}\) (position is given in meters). Restart.

    1. What is the tension in each string?
    2. What is the force of each ringstand on the crossbar?

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{3}\): Calculate the normal forces on the front and rear tires of a truck

    The side view of a truck on a level road is shown in the animation. The mass of the truck is \(1230\text{ kg}\). Suppose the resultant force of the road on the front set of tires is \(4000\text{ N}\), in the upward direction of course (position is given in meters). Restart.

    1. What is the resultant force of the road on the rear set of tires?
    2. What is the horizontal distance from the front axle to the center of mass of the truck?

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{4}\): Analyze a box pulled by a rope

    A \(10\text{-kg}\) box sits on a table as shown in the animation. A rope is attached to the box, and you pull the rope to the right with a certain force. Assume that the coefficient of static friction is great enough so that the box doesn't slip first (position is given in meters). Restart.

    1. What minimum force will make the box tip?
    2. What is the minimum value of the coefficient of static friction for the box NOT to slide before tipping?
    3. If you would like to apply a greater force to the box, yet minimize the risk of tipping, what should you do?

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{5}\): Analyze a bicycle wheel suspended on the edge of a curb

    The rear wheel of a bicycle is in equilibrium as it rests against the corner of a curb as shown in the animation. For purposes of this problem, neglect the force of the bicycle frame on the wheel. The wheel is not touching the ground, and its mass is \(0.40\text{ kg}\). Neglect the mass of the bicycle chain (position is given in meters). Restart.

    1. Assuming that the top and bottom chain tensions are equal, what is the tension in the chain?
    2. What is the magnitude and direction of the force of the curb on the wheel?

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{6}\): Analyze a beam attached to a wall and wire

    A string is attached between the end of a narrow uniform beam and the wall. Friction on the beam keeps the left side from falling. The magnitude of the weight of the beam is \(w\) (position is given in meters). Restart.

    1. What is the ratio of the tension in the string to the weight, \(T/w\)?
    2. What is the magnitude and direction of the force of the wall on the beam?
    3. What is the minimum coefficient of static friction required for the left end of the beam not to slip?

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{7}\): Analyze a system of two rods, each attached to hinges

    Two uniform rods are hinged to a wall and screwed to each other as shown in the animation. The mass of the blue rod is \(2.0\text{ kg}\) and the mass of the black rod is \(1.5\text{ kg}\) (position is given in meters). Restart.

    1. What are the magnitude and direction of the force of the top hinge on the black rod?
    2. What are the magnitude and direction of the force of the bottom hinge on the blue rod?
    3. What are the magnitude and direction of the force on each rod due to the screw attaching the rods? Your answer should include two forces, the force of the screw on the blue rod and the force of the screw on the black rod.

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{8}\): Apply equilibrium conditions to a see-saw

    A seesaw, usually in equilibrium, is shown in the animation. A little girl named Melody sits on the seesaw and thereby applies a force of \(200\text{ N}\) to the seesaw as indicated by the red vector labeled \(F_{M}\). The seesaw weighs \(500\text{ N}\) (position is given in meters). Restart.

    1. Where should her dad sit on the seesaw in order to keep the system in equilibrium if he applies a bulky \(900\text{ N}\) to the seesaw when sitting on it?
    2. What is the force of the axle on the seesaw, assuming that the only other force applied to the seesaw besides Melody and her dad is the force of the axle?
    3. When Melody's dad wants to practice a circus act by applying a \(900\text{ N}\) force to the far right edge of the seesaw, Melody's mom immediately rushes to her aid by applying a downward force on the seesaw to keep the system in equilibrium. If the force of Mom on the seesaw has a magnitude of \(750\text{ N}\), at what location on the seesaw is it applied?

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{9}\): Analyze a pendulum held at some angle in equilibrium

    A long pendulum is made of a pendulum bob of mass \(10.0\text{ kg}\) attached to a lightweight cable. By pushing horizontally on the pendulum bob, you keep it in equilibrium. Note: The hand is not drawn to scale; it appears larger than its actual dimensions (position is given in meters). Restart.

    1. What is the force of your hand on the pendulum bob?
    2. What is the force of the cable on the pendulum bob (i.e., tension)?
    3. Suppose you wish to make it easier on yourself by applying the minimum force necessary to keep the pendulum bob in equilibrium, while maintaining its position as shown in the animation. In this case, what are the magnitude and direction of the minimum force of your hand on the pendulum bob, and what would then be the tension in the cable?

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{10}\): Calculate reaction forces and torques on a stuck lever

    An emergency lever (the red, sideways, L-shaped object) is designed to rotate clockwise about an axle in a hinge (the gray half circle) as shown in the animation. Assume the lever is rigid, is made of uniform material, and has a mass of \(0.20\text{ kg}\). The axle (the black circle) exerts a frictional force on the lever (position is given in meters). Restart.

    1. What are the magnitude and direction of the torque on the lever due to friction between the lever and axle?
    2. What are the magnitude and direction of the net force of the axle on the lever?
    3. In trying to turn the lever, you apply a force of magnitude \(5.0\text{ N}\) in the direction shown in the following animation. Show Animation with Applied Force Vector. But the lever remains in equilibrium. What is the torque on the lever due to friction between the lever and axle, and what are the magnitude and direction of the net force of the axle on the lever?
    4. Suppose you keep the magnitude of the applied force the same (\(5.0\text{ N}\)) but wish to apply a greater torque to the lever. What could you do differently?
    5. Suppose the maximum torque on the lever due to friction at the axle is \(10.0\text{ N}\cdot\text{m}\). You decide to apply a force straight downward at the left end of the lever. What is the minimum value of the magnitude of the force in order to just barely turn the lever?

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{11}\): Determine where the normal force acts on a box on a ramp

    A wood block of uniformly distributed mass sits in equilibrium on a ramp as shown in the animation. Its mass is \(0.20\text{ kg}\) (position is given in meters). Restart.

    1. If you replace the "load" of the ramp on the bottom surface of the block with a single normal force acting at a certain distance from the front edge of the block, what are the magnitude and direction of this force and at what location does the force act?
    2. What is the force of static friction on the block?

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{12}\): Calculate center of mass

    Four spheres are shown in the animation. A blue sphere is half as massive as a red one, and a purple sphere is twice as massive as a red one. Where should the purple one be placed in order for the center of gravity to be at the location of the black dot (position is given in meters)? Restart.

    Problem authored by Aaron Titus.

    Exercise \(\PageIndex{13}\): Determine the mass of the board

    A \(2\text{-kg}\) box sits \(0.3\text{ m}\) from the right end of a board of unknown weight. Two supports exert forces on the board (position is given in meters). You can drag the second support from the left to the right to view how that force changes with its position. The arrows represent the relative sizes of the force vectors for the movable support and the box, but their length does not represent their actual magnitudes (the actual value of the forces, as well as the separation between the supports, is shown in the table). The board is \(6\text{ m}\) long and support 1 is \(0.3\text{ m}\) in from the left edge. Determine the mass of the board. Restart.

    Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.

    This page titled 11.1.13.3: Problems is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by Wolfgang Christian, Mario Belloni, Anne Cox, Melissa H. Dancy, and Aaron Titus, & Thomas M. Colbert.