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A spring is stretched 175 mm by an 8-kg block. If the block is displaced 100 mm downward from its equilibrium position and given a downward velocity of 1.50 m/s, determine the differential equation which describes the motion. Assume that positive displacement is downward. Also, determine the position of the block when t = 0.22 s.
The differential equation which describes the motion and the position of the block when
Answer to Problem 1P
The differential equation which describes the motion is
Explanation of Solution
Given:
The spring stretched,
The mass of the block,
Distance moved below from equilibrium position,
The downward velocity of block,
The given time,
Show the free body diagram of theblock as in Figure (1).
Conclusion:
Refer Figure (1),
Resolve forces along
Here, acceleration due to gravity is
Substitute
Express the natural frequency.
Substitute
Substitute
Hence, the differential equation which describes the motion is
Write the solution in Equation (IV) in the form of differential equation.
Here, constants are
Differentiate Equation (V) with respect to time to get,
Write the boundary conditions:
At
The conditions are:
Apply the boundary conditions in Equation (V),
Apply the boundary conditions in Equation (VI),
Substitute
Substitute
Hence, the position of the block when
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Chapter 22 Solutions
Engineering Mechanics: Dynamics (14th Edition)
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