A spring has a relaxed length of 34 cm (0.34 m) and its spring stiffness is 10 N

Question

A spring has a relaxed length of 34 cm (0.34 m) and its spring stiffness is 10 N/m. You glue a 65 gram block (0.065 kg) to the top of the spring, and push the block down, compressing the spring so its total length is 15 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 0.12-second interval after you release the block, by applying the Momentum Principle in three steps each of 0.04-secondduration.

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 = 0.04 seconds, what will the new momentum and velocity of the block be?

Position update: Initially the bottom of the block is at y = 0.15 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 = 0.04 seconds?

y =

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 × 0.04 = 0.08 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 × 0.04 = 0.08 seconds?

y =

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 × 0.04 = 0.12 seconds,

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

vy=

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 × 0.04 = 0.12 seconds?

y =

Fspring, y = Relaxed length Push down, release from rest

Explanation / Answer

Step 1:

force exerted by the spring =spring constant*compression

=10*(0.34-0.15)=1.9 N

force is directed vertically upwards.

part 2:

force by earth=weight of the block=mass*g

=0.065*9.8=0.637 N

force is directed vertically downward.

part 3:

net force is in vertically upward direction.

net force = 1.9-0.637=1.263 N

part 4:

change in momentum=force*time duration

in py=1.263*0.04=0.05052 kg.m/s

part 5:

velocity at t=0.04 seconds=py/mass=0.05052/0.065=0.7772 m/s

part 6:

new position at t=0.04 seconds=initial position+ velocity*time

=0.15+0.7772*0.04=0.181088 m

Step 2:

force exerted by the spring =spring constant*compression

=10*(0.34-0.181088)=1.58912 N

force is directed vertically upwards.

part 2:

force by earth=weight of the block=mass*g

=0.065*9.8=0.637 N

force is directed vertically downward.

part 3:

net force is in vertically upward direction.

net force = 1.58912-0.637=0.95212 N

part 4:

change in momentum=force*time duration

==> py-0.05052=0.95212*0.04

==>py=0.0886 kg.m/s

part 5:

velocity at t=0.04 seconds=py/mass=0.0886/0.065=1.36315 m/s

part 6:

new position at t=0.04 seconds=initial position+ velocity*time

=0.181088+1.361315*0.04=0.2356 m

Step 3:

force exerted by the spring =spring constant*compression

=10*(0.34-0.2356)=1.044 N

force is directed vertically upwards.

part 2:

force by earth=weight of the block=mass*g

=0.065*9.8=0.637 N

force is directed vertically downward.

part 3:

net force is in vertically upward direction.

net force = 1.044-0.637=0.407 N

part 4:

change in momentum=force*time duration

==> py-0.0886=0.407*0.04

==>py=0.10488 kg.m/s

part 5:

velocity at t=0.04 seconds=py/mass=0.10488/0.065=1.61354 m/s

part 6:

new position at t=0.04 seconds=initial position+ velocity*time

=0.2356+1.61354*0.04=0.30014 m

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