Use the exact values you enter to make later calculations. A spring has a relaxe

Question

Use the exact values you enter to make later calculations.

A spring has a relaxed length of 27 cm (0.27 m) and its spring stiffness is 19 N/m. You glue a 77 gram block (0.077 kg) to the top of the spring, and push the block down, compressing the spring so its total length is 18 cm. You make sure the block is at rest, then at time you quickly move your hand away. The block begins to move upward, because the upward force on the block by the spring is greater than the downward force on the block by the Earth. Calculate vs. time for the block during a 0.06-second interval after you release the block, by applying the Momentum Principle in three steps each of 0.02-second duration.

We will only consider the components in the following calculations, because there is no change in or

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Momentum update: Just after releasing the block, the momentum of the block is zero. Calculate the average net force during the next time interval by the force you just calculated. At 0.02 seconds what will the new momentum and velocity of the block be?

Position update: Initially the bottom of the block is at 0.18 m. Calculating the average velocity in the first time interval by the final velocity, what will be the new position of the bottom of the block at time 0.02 seconds?

Momentum update: Calculate the average net force during the next time interval by the force you just calculated. At time 2 0.02 = 0.04 seconds, what will the new momentum and velocity of the block be?

Position update: Calculating the average velocity in the second time interval by the final velocity, what will be the new position of the bottom of the block at time 2 0.02 = 0.04 seconds?

Momentum update: Calculate the average net force during the next time interval by the force you just calculated. At time 3 0.02 = 0.06 seconds, what will the new momentum and velocity of the block be?

Position update: Calculating the average velocity in the second time interval by the final velocity, what will be the new position of the bottom of the block at time 3 0.02 = 0.06 seconds?

Applying the Momentum Principle in this way to predict motion is a "numerical integration"—adding up the cumulative effects of many impulses in a succession of time intervals. As you can see, this can be very tedious if done by hand, and this task is much more easily carried out by programming a computer to do all the repetitive operations. However, it is important to do some calculations by hand to understand in detail the procedure that you would program a computer to carry out

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Explanation / Answer

given, relaxed length of spring, Lo = 27 cm = 0.27 m

spring stiffness, k = 19 N/m

mass of block, m = 0.077 kg

intiial length, l = 0.18 m

1. jUST AFTER RELEASING THE BLOCK

force exerted on the block by the spring, F = k(0.27 - 0.18) = 1.71 N

force exerted by block on the earth = force exerted by earth on the block = mg = 0.75537 N

net force = 1.71 - 0.75537 = 0.95463 N

2. force at t = 0, F = 0.95463 N ( net force)

assuming tihs to be conjstant for the next 0.02 s

change in momentum = F*dt = 0.95463*0.002 = 0.00190926 kg m/s

velocity of block = v

mv = 0.00190926

v = 0.0247 m/s

3. assuming constant velocity for the first 0.003 s

v = 0.0247 m/s

t = 0.02 s

so0 new momentum = mv = 0.0019019 kg m/s

new velocity = v'

v' = momentum/mass = 0.0247 m/s

4. at x = 0.02 m

force exerted on the block by the spring, F = k(0.27 - 0.20) = 1.33N

force exerted by block on the earth = force exerted by earth on the block = mg = 0.75537 N.

net force on the block = 0.58N

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