A spring has a relaxed length of 32 cm (0.32 m) and its spring stiffness is 9 N/

Question

A spring has a relaxed length of 32 cm (0.32 m) and its spring stiffness is 9 N/m. You glue a 73 gram block (0.073 kg) to the top of the spring, and push the block down, compressing the spring so its total length is 13 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 approximately y vs. time for the block during a 0.24-second interval after you release the block, by applying the Momentum Principle in three steps each of 0.08-second duration.

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Step 1

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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: Fspring,y = ? FEarth,y = ? Fnet,y = ?

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Momentum update: Just after releasing the block, the momentum of the block is zero. Approximate the average net force during the next time interval by the force you just calculated. At t = 0.08 seconds, what will the new momentum and velocity of the block be? py = vy =

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Position update: Initially the bottom of the block is at y = 0.13 m. Approximating 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 = 0.08 seconds? y =

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Step 2

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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: Fspring,y = N FEarth,y = N Fnet,y = N

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Momentum update: Approximate the average net force during the next time interval by the force you just calculated. At time t = 2

Explanation / Answer

In the answer given below, acceleration due to gravity is taken as 9.8 m/s^2 . And the answers are given accordingly. The answers have also been rounded off approximately. If you want the exact answers, you can follow the same method, use no approximation and calculate more accurately. Step1: just after release, force exerted by spring = stiffness*compression = 9*(0.32-0.13) = 9*0.19 = 1.71N upwards force exerted on block by earth = mg = 0.073*9.8 = 0.71 N downwards Therefore, net force on block = 1.71 - 0.71 = 1N upwards. Momentum update: So we consider this force to be acting on the block for the first 0.08s Therefore, change in momentum = force*time = 1*0.08 = 0.08 kgm/s final momentum - initial momentum = 0.08 kgm/s Therefore, momentum after t = 0.08s = 0.08kgm/s upwards velocity = momentum/mass = 0.08/0.073 = 1.1 m/s (approx.) Position update: the block travels the first 0.08 s with velocity 1.1 m/s upwards therefore, y new = 0.13 + (0.08*1.1) = 0.13 + 0.088 = 0.218 m Step 2: force on block due to spring = 9*(0.32 - 0.218) = 9*(0.102) = 0.918 N upwards force exerted on block by earth = mg = 0.073*9.8 = 0.71 N downwards Therefore, net force on block = 0.918 - 0.71 = 0.208N upwards. Momentum update: So we consider this force to be acting on the block for the next 0.08s Therefore, change in momentum = force*time = 0.208*0.08 = 0.0166 kgm/s final momentum - initial momentum = 0.0166 kgm/s Therefore, momentum after t = 0.16s = 0.08 + 0.0166 = 0.0966 kgm/s upwards velocity = momentum/mass = 0.0966/0.073 = 1.323 m/s (approx.) Position update: the block travels the next 0.08 s with velocity 1.323 m/s therefore, y new = 0.218 + 1.323*0.08 = 0.218 + 0.106 = 0.324m Step 3: force on block due to spring = 9*(0.32 - 0.324) = 9*(-0.004) = 0.036 N downwards force exerted on block by earth = mg = 0.073*9.8 = 0.71 N downwards Therefore, net force on block = 0.036 + 0.71 = 0.746N downwards. Momentum update: So we consider this force to be acting on the block for the next 0.08s Therefore, change in momentum = force*time = -0.746*0.08 = -0.06 kgm/s final momentum - initial momentum = -0.06 kgm/s Therefore, momentum after t = 0.24s = 0.0966 - 0.06 = 0.0366 kgm/s upwards velocity = momentum/mass = 0.0366/0.073 = 0.5 m/s (approx.) Position update: the block travels the next 0.08 s with velocity 0.5 m/s therefore, y new = 0.324 + 0.08*0.5 = 0.364m

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