An object in simple harmonic motion has an amplitude of 8.0 cm, n angular frequency of 0.25 rad/s, and a phase constant of π rad. Draw a velocity graph showing two cycles of the motion.

Verified step by step guidance

1. Step 1: Recall the velocity equation for simple harmonic motion, which is given by v(t) = -ωA sin(ωt + φ), where ω is the angular frequency, A is the amplitude, and φ is the phase constant.
2. Step 2: Substitute the given values into the velocity equation. Here, ω = 0.25 rad/s, A = 8.0 cm (convert to meters if needed), and φ = π rad. The equation becomes v(t) = -0.25 × 8.0 × sin(0.25t + π).
3. Step 3: Analyze the phase constant φ = π rad. This means the sine function starts at its minimum value because sin(π) = 0. The velocity graph will be shifted accordingly.
4. Step 4: Determine the period of the motion using the formula T = 2π/ω. Substituting ω = 0.25 rad/s, we find T = 2π/0.25 = 8π seconds. Two cycles of motion will span 16π seconds.
5. Step 5: Plot the velocity graph over the time interval [0, 16π] using the equation v(t) = -0.25 × 8.0 × sin(0.25t + π). The graph will be sinusoidal, with peaks and troughs corresponding to the maximum and minimum velocities, and a period of 8π seconds.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The motion is characterized by a restoring force proportional to the displacement from the equilibrium, leading to sinusoidal motion. Key parameters include amplitude, frequency, and phase constant, which define the motion's characteristics.

Velocity in SHM

In Simple Harmonic Motion, the velocity of the oscillating object varies sinusoidally with time. It can be expressed as v(t) = -Aω sin(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant. The velocity reaches its maximum at the equilibrium position and is zero at the maximum displacement (amplitude).

Graphing SHM

Graphing the velocity of an object in Simple Harmonic Motion involves plotting velocity against time. The resulting graph is a sine wave, reflecting the periodic nature of the motion. The amplitude of the velocity graph corresponds to the maximum speed of the object, while the frequency of the wave relates to the angular frequency of the motion.