A block of unknown mass is attached to a spring with a spring constant of 7.4 N/m and undergoes simple harmonic motion with an amplitude of 5.5 cm. When the mass is halfway between its equilibrium position and the endpoint, its speed is measured to be 38.8 cm/s. Calculate the mass of the block.

The mass of the block undergoing simple harmonic motion with a spring constant of 7.4 N/m and a measured speed at half amplitude is approximately 49.27 g.

Explanation:

To calculate the mass of the block undergoing simple harmonic motion (SHM), we can use the conservation of energy principle which states that the total mechanical energy in SHM is constant and is composed of kinetic and potential energy. The spring constant k is given as 7.4 N/m, and the speed v of the block when it is halfway (at an amplitude of 2.75 cm or 0.0275 m) is 38.8 cm/s or 0.388 m/s.

At the halfway point, the total mechanical energy E of the system is the sum of kinetic energy (KE) and potential energy (PE) stored in the spring:

KE = ½mv²
PE = ½kx²

Since no energy is lost, E is also the maximum potential energy when the spring is at its full amplitude of 5.5 cm or 0.055 m, so:

E = ½kA²

Equating the two expressions for E:

½kA² = ½kx² + ½mv²

Solving for m, we get:

m = ½k(A² - x²) / v²

Substituting the given values:

m = ½×7.4(0.055² - 0.0275²) / 0.388²

m = ½×7.4(0.003025 - 0.00075625) / 0.150544

m = 0.7421 / 0.150544

m ≈ 0.04927 kg or 49.27 g

The mass of the block is approximately 49.27 g.