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 34 cm (0.34 m) and its spring stiffness is 9 N/m. You glue a 71 gram block (0.071 kg) to the top of the spring, and push the block down, compressing the spring so its total length is 12 cm. You make sure the block is at rest, then at time

t = 0

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 y vs. time for the block during a 0.18-second interval after you release the block, by applying the Momentum Principle in three steps each of 0.06-second duration.

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

STEP 1

Force: Just after releasing the block, calculate the force exerted on the block by the spring, the force exerted on the block by the Earth, and the net force:

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

t = 0.06 seconds,

what will the new momentum and velocity of the block be?

Position update: Initially the bottom of the block is at

y = 0.12 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

t = 0.06 seconds?

y =  m

STEP 2

Force: At the new position, calculate the force exerted on the block by the spring, the force exerted on the block by the Earth, and the net force (remember near the Earth's surface, the gravitational force due to the Earth is very nearly constant):

Momentum update: Calculate the average net force during the next time interval by the force you just calculated. At time

t = 2 × 0.06 = 0.12 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

t = 2 × 0.06 = 0.12 seconds?

y =  m

STEP 3

Force: At the new position, calculate the force exerted on the block by the spring, the force exerted on the block by the Earth, and the net force (remember near the Earth's surface, the gravitational force due to the Earth is very nearly constant):

Momentum update: Calculate the average net force during the next time interval by the force you just calculated. At time

t = 3 × 0.06 = 0.18 seconds,

what will the new momentum and velocity of the block be?

Position update: Calculating the average velocity in the third time interval by the final velocity, what will be the new position of the bottom of the block at time

t = 3 × 0.06 = 0.18 seconds?

y =  m

Fspring, y = N FEarth, y = N Fnet, y = N Relaxed length Push down, release from rest

Explanation / Answer

Step1:
just after release, force exerted by spring = stiffness*compression

= 9*(0.34-0.12) = 1.98N upwards

force exerted on block by earth = mg
= 0.071*9.8 = 0.6958 N downwards

Therefore, net force on block = 1.98 - 0.6958 =1.2842 upwards.

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Momentum update:

So we consider this force to be acting on the block for the first 0.06s

Therefore, change in momentum = force*time = 1.2842*0.06 = 0.077 kgm/s

velocity = momentum/mass = 0.077/0.071 = 1.085 m/s

Position update:

the block travels the first 0.06 s with velocity 1.085 m/s upwards
therefore, y new = 0.12 + (0.06*1.085) = 0.13 + 0.088 = 0.1851 m

----------------------------------------------------------------------------------------------

Step 3:

force on block due to spring =

9*(0.12 - 0.1851) =0.5859 N downwards

force exerted on block by earth = mg

= 0.071*9.8 = 0.6958 N downwards
Therefore, net force on block = 0.5859 +0.6958 = 1.2817N downwards.


Momentum update:

So we consider this force to be acting on the block for the next 0.06s

Therefore, change in momentum = force*time = -1.2817*0.06 = -0.076 kgm/s

final momentum - initial momentum = -0.076 kgm/s

Therefore, momentum after t = 0.18s = 0.077 - 0.076 = 0.001kgm/s upwards
velocity = momentum/mass = 0.01/0.071 = 0.014 m/s (approx.)

Position update:

the block travels the next 0.06 s with velocity 0.014 m/s
therefore, y new = 0.1851 + 0.06*0.014 = 0.1859m

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