Calculate the mass of the block, the period of the motion, and the maximum speed of the block.   A block of unknown mass is attached to a spring with a spring constant of 10 N/m and undergoes...

Calculate the mass of the block, the period of the motion, and the maximum speed of the block.

 

A block of unknown mass is attached to a spring with a spring constant of 10 N/m and undergoes simple harmonic motion with an amplitude of 8.0 cm. When the block is 1/4 of the way between its equilibrium position and the endpoint, its speed is measured to be 30.0 cm/s.

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mathlete13 | High School Teacher | (Level 2) Adjunct Educator

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To calculate the block's mass, first calculate the energy of the system using the formula `E=1/2kA^2` , where E is the energy in Joules, k is the spring constant in N/m, and A is the amplitude in m. Converting 8 cm into meters gives 0.08 m for the amplitude. So

`E=1/2kA^2=1/2(10)(0.08^2)=0.032 J`

Since the total energy of the system is the sum of its kinetic and potential energy and its speed is measured at 1/4 of its amplitude, we also know that `E=1/2k(A/4)^2 + 1/2mv^2` , where m is the block's mass in kg and v is its velocity in m/s. To find m, substitute in the known values and solve for the unknown variable.

`0.032=1/2(10)(0.08/4)^2+1/2m(0.3^2)`

`0.032=0.002+0.045m`

`0.03=0.045m`

`m~~0.667`

So the block's mass is approximately 0.667 kg, or 667 g.

To find the period of the motion, use the formula `T=2pisqrt(m/k)` , where T is the period in seconds. Plugging in the known values gives

`T=2pisqrt(0.667/10)~~1.62`

Therefore, the period is 1.62 s.

Finally, the block's maximum velocity is found using the formula `v_(max)=(2piA)/T`, so

 `v_(max)=(2pi*0.08)/1.62~~0.31`

The block's maximum velocity is 0.31 m/s, or 31 cm/s.

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