Final answer:
To determine the equations of motion for a block on a spring undergoing Simple Harmonic Motion, apply Hooke's Law, find the position function from SHM properties, and use calculus to determine the velocity and acceleration as functions of time.
Step-by-step explanation:
When a block attached to a spring is pulled and released, it exhibits Simple Harmonic Motion (SHM). The block has its maximum potential energy and zero kinetic energy at the moment of release. As it passes through the equilibrium position (x=0), its velocity is at the maximum, and the kinetic energy is highest. The equations of motion for a block on a spring can be determined using Hooke's Law and the properties of SHM.
Step-by-step Calculation
- Use Hooke's Law, F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from equilibrium.
- For SHM, the position x as a function of time t is given by x(t) = A * cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant.
- From the period T, calculate the angular frequency ω using ω = 2π/T.
- Determine the amplitude A from the maximum displacement provided.
- Since the block is released from rest, the phase constant φ is zero.
- The velocity can be found by differentiating the position function, v(t) = -Aω * sin(ωt).
- The acceleration is the second derivative of the position function, a(t) = -Aω2 * cos(ωt).
The potential energy at any position is given by U = 1/2 * kx2, and the kinetic energy can be calculated using KE = 1/2 * mv2, where m is the mass of the block.