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.
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
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.
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
Therefore, the period is 1.62 s.
Finally, the block's maximum velocity is found using the formula `v_(max)=(2piA)/T`, so
The block's maximum velocity is 0.31 m/s, or 31 cm/s.