What is the motion of the block when released from a compressed spring?

In summary, the problem involves a 15.8-in. spring compressed to 7.1-in. and released, causing a sliding block A to accelerate with an initial value of 160 ft/sec2. The acceleration decreases with x-movement until the spring regains its original length. The time t is calculated for the block to move 4.35 in. and 8.7 in. The equations used are a = 160-220X, v^2 = 320x-220x^2, and V = dx/dt. The integral of dx/sqrt(320x-220x^2) is used to solve for dt and integrate for T, but the process may be incorrect. The motion
  • #1
allyfranken
4
0

Homework Statement



The 15.8-in. spring is compressed to a 7.1-in. length, where it is released from rest and accelerates the sliding block A. The acceleration has an initial value of 160 ft/sec2 and then decreases linearly with the x-movement of the block, reaching zero when the spring regains its original 15.8-in. length. Calculate the time t for the block to go (a) 4.35 in. and (b) 8.7 in.

Homework Equations



a = 160 - kx

vdV = adX
V = dx/dt

The Attempt at a Solution



k = 160/((15.8-7.1)/12)

a = 160 - 220X

Then i integrated vDV = 160 - 220x
and got: v^2 = 320x - 220x^2
solved for V = sqrt(320x - 220x^2)

now I know that V = dx/dt and to solve for dt and integrate for T. However I get stuck at the integral of dx/sqrt(320x - 220x^2) so I assume I am doing something wrong in the process of getting there.
 
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  • #2
If the block were attached to the end of the spring and the system allowed to continue to move, what kind of motion would you see? What does that suggest for the form of x as a function of t?
 

FAQ: What is the motion of the block when released from a compressed spring?

What is spring motion dynamics?

Spring motion dynamics is the study of the movement and behavior of objects attached to a spring as it undergoes oscillatory motion.

How does Hooke's law relate to spring motion dynamics?

Hooke's law states that the force exerted by a spring is directly proportional to the displacement of the object attached to it. This is a fundamental principle in spring motion dynamics.

What factors affect the frequency of a spring's oscillations?

The frequency of a spring's oscillations is affected by the mass of the object attached to it, the stiffness of the spring, and the initial displacement of the object.

How does damping affect the motion of a spring?

Damping is the force that opposes the motion of an oscillating spring. It can decrease the amplitude of the oscillations and also affect the frequency of the motion.

What real-life applications involve spring motion dynamics?

Spring motion dynamics is used in numerous real-life applications, such as in car suspensions, shock absorbers, and door closers. It is also important in understanding the behavior of structures during earthquakes and other vibrations.

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