ENGR.MECH.: DYNAMICS-EBOOK>I<
ENGR.MECH.: DYNAMICS-EBOOK>I<
14th Edition
ISBN: 9781292088785
Author: HIBBELER
Publisher: INTER PEAR
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Chapter 22.1, Problem 1P

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.

Expert Solution & Answer
Check Mark
To determine

The differential equation which describes the motion and the position of the block when t=0.22s .

Answer to Problem 1P

The differential equation which describes the motion is y¨+56.1y=0 and the position of the block when t=0.22s is 0.192m .

Explanation of Solution

Given:

The spring stretched, x=175mmor(0.175m) .

The mass of the block, m=8kg .

Distance moved below from equilibrium position, y=100mmor(0.1m) .

The downward velocity of block, v=1.50m/s .

The given time, t=0.22s .

Show the free body diagram of theblock as in Figure (1).

ENGR.MECH.: DYNAMICS-EBOOK>I<, Chapter 22.1, Problem 1P

Conclusion:

Refer Figure (1),

Resolve forces along y direction.

+Fy=maymgk(y+yst)=my¨mgkykyst=my¨ (I)

Here, acceleration due to gravity is g and spring constant is k .

Substitute mg for kyst in Equation (I).

mgkymg=my¨y¨+kmy=0y¨+(ωn)2y=0 (II)

Express the natural frequency.

ωn=km=mg/xm (III)

Substitute 8kg for m , 9.81m/s2 for g and 0.175m for x in Equation (III).

ωn=8(9.81)/0.1758=56.057=7.487

Substitute 7.487 for ωn in Equation (II).

y¨+(7.487)2y=0y¨+56.1y=0 (IV)

Hence, the differential equation which describes the motion is y¨+56.1y=0 .

Write the solution in Equation (IV) in the form of differential equation.

y=Asinωnt+Bcosωnt (V)

Here, constants are AandB respectively.

Differentiate Equation (V) with respect to time to get,

y˙=v=AωncosωntBωnsinωnt (VI)

Write the boundary conditions:

At

t=0 :

The conditions are:

y=0.1mv=v0=1.50m/s

Apply the boundary conditions in Equation (V),

0.1=Asin0+Bcos0B=0.1m

Apply the boundary conditions in Equation (VI),

v0=Aωncos00A=v0ωn (VII)

Substitute 1.50m/s for v0 and 7.487 for ωn in Equation (VII).

A=1.507.487=0.2003m

Substitute 0.2003m for A , 7.487 for ωn , 0.22s for t and 0.1m for B in Equation (V).

y=(0.2003)sin[7.487(0.22)]+0.1cos[7.487(0.22)]=0.19971+(0.00763)=0.192m

Hence, the position of the block when t=0.22s is 0.192m .

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Chapter 22 Solutions

ENGR.MECH.: DYNAMICS-EBOOK>I<

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