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A spring is stretched mm by an 8-kg block. If the block is displaced mm downward from its equilibrium position and given a downward velocity of 1. Assume that positive displacement is downward. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. A spring is stretched mm by a kg block. If the block is displaced mm downward from its equilibrium position and given a downward.

What is the phase angle? When a lb weight is suspended from a spring, the spring is stretched a distance of 4 in. Determine the natural frequency and the period of vibration for a lb weight attached to the same spring. When a 3-kg block is suspended from a spring, the spring is stretched a distance of 60 mm.

Determine the natural frequency and the period of vibration for a 0. If the block is given an upward velocity of 0. Assume that positive displacement is measured downward. If the weight is pushed 1 in. What is the amplitude and the natural frequency of the vibration?

Assume positive displacement is downward. If the block is pushed 50 mm upward from its equilibrium position and then released from rest, determine the equation that describes the motion. What are the amplitude and the frequency of the vibration? The uniform rod of mass m is supported by a pin at A and a spring at B. If B is given a small sideward displacement and released, determine the natural period of vibration. Solution 1 Equation of Motion.

While standing in an elevator, the man holds a pendulum which consists of an in. Determine the natural period of vibration of the uniform bar of mass m when it is displaced downward slightly and released. The body of arbitrary shape has a mass m, mass center at G, and a radius of gyration about G of kG.

If it is displaced a slight amount u from its equilibrium position and released, determine the natural period of vibration. Neglect the mass of the rod. A platform, having an unknown mass, is supported by four springs, each having the same stiffness k. When nothing is on the platform, the period of vertical vibration is measured as 2. Determine the mass of a block placed on the empty platform which causes the platform to vibrate vertically with a period of 5.

What is the stiffness k of each of the springs? A block of mass m is suspended from two springs having a stiffness of k1 and k2, arranged a parallel to each other, and b as a series. Determine the equivalent stiffness of a single spring with the same oscillation characteristics and the period of oscillation for each case.

The kg block is suspended from two springs having a different stiffness and arranged a parallel to each other, and b as a series. If the natural periods of oscillation of the parallel system and series system are observed to be 0. The uniform beam is supported at its ends by two springs A and B, each having the same stiffness k. When nothing is supported on the beam, it has a period of vertical vibration of 0.

If a kg mass is placed at its center, the period of vertical vibration is 1. Compute the stiffness of each spring and the mass of the beam. The slender rod has a mass of 0. The period of vibration of the rod can be set by fixing the 0. Neglect the size of the collar. A uniform board is supported on two wheels which rotate in opposite directions at a constant angular speed.

If the coefficient of kinetic friction between the wheels and board is m, determine the frequency of vibration of the board if it is displaced slightly, a distance x from the midpoint between the wheels, and released. If the wire AB is subjected to a tension of 20 lb, determine the equation which describes the motion when the 5-lb weight is displaced 2 in. The bar has a length l and mass m. It is supported at its ends by rollers of negligible mass.

If it is given a small displacement and released, determine the natural frequency of vibration. The kg disk, is pinned at its mass center O and supports the 4-kg block A.

If the belt which passes over the disk is not allowed to slip at its contacting surface, determine the natural period of vibration of the system. Referring to the FBD and kinetic diagram of the system, Fig. Substitute this result into Eq. The kg disk is pin connected at its mass center. Determine the natural period of vibration of the disk if the springs have sufficient tension in them to prevent the cord from slipping on the disk as it oscillates.

Hint: Assume that the initial stretch in each spring is dO. Solution Equation of Motion. Referring to the FBD of the disk, Fig.

If the disk in Prob. If the flywheel is given a small angular displacement of u and released, determine the natural period of oscillation.

Referring to Fig. The 6-lb weight is attached to the rods of negligible mass. Determine the natural frequency of vibration of the weight when it is displaced slightly from the equilibrium position and released. If a car, having a mass of 1. Determine the moment of inertia of the car about an axis passing through G2. SOLUTION Free-body Diagram: When an object arbitrary shape having a mass m is pinned at O and being displaced by an angular displacement of u, the tangential component of its weight will create the restoring moment about point O.

Substituting these values into Eq. The plate of mass m is supported by three symmetrically placed cords of length l as shown. If the plate is given a slight rotation about a vertical axis through its center and released, determine the natural period of oscillation. Determine the differential equation of motion of the 3-kg block when it is displaced slightly and released.

The surface is smooth and the springs are originally unstretched. Determine the natural period of vibration of the pendulum.

If the kg wheel is displaced a small amount and released, determine the natural period of vibration. The wheel rolls without slipping. Solution Energy Equation. Determine the differential equation of motion of the 3-kg spool. Assume that it does not slip at the surface of contact as it oscillates.

Determine the natural period of vibration of the 3-kg sphere. Neglect the mass of the rod and the size of the sphere. If the lower end of the 6-kg slender rod is displaced a small amount and released from rest, determine the natural frequency of vibration.

Energy Equation. Thus, the Kinetic energy is 3 3. The disk has a weight of 30 lb and rolls without slipping on the horizontal surface as it oscillates about its equilibrium position. If the disk is displaced, by rolling it counterclockwise 0. Since the disk rolls without 2 Time Derivative. Taking the time derivative of the above equation, 0.

The machine has a mass m and is uniformly supported by four springs, each having a stiffness k. Determine the natural period of vertical vibration.

If it is supported in the horizontal plane by a ball-and-socket joint at A and a cable at B, determine the natural frequency of vibration when the end B is given a small horizontal displacement and then released. If the slender rod has a weight of 5 lb, determine the natural frequency of vibration. The springs are originally unstretched.

When the rod undergoes a small angular displacement u, its mass center is 0. Time Derivative: Taking the time derivative of Eq. From Eq. Applying Eq. What is the general solution of this equation?

C cos vt. A block which has a mass m is suspended from a spring having a stiffness k.

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