In a simple harmonic motion, if the displacement is given by x(t) = A cos(ωt + φ), what is the phase constant φ?
Practice Questions
1 question
Q1
In a simple harmonic motion, if the displacement is given by x(t) = A cos(ωt + φ), what is the phase constant φ?
0
π/2
π
Depends on initial conditions
The phase constant φ depends on the initial conditions of the motion, such as the initial position and velocity.
Questions & Step-by-step Solutions
1 item
Q
Q: In a simple harmonic motion, if the displacement is given by x(t) = A cos(ωt + φ), what is the phase constant φ?
Solution: The phase constant φ depends on the initial conditions of the motion, such as the initial position and velocity.
Steps: 7
Step 1: Understand the equation x(t) = A cos(ωt + φ). Here, x(t) represents the displacement at time t, A is the amplitude, ω is the angular frequency, and φ is the phase constant.
Step 2: Recognize that the phase constant φ determines the starting position of the motion at time t = 0.
Step 3: To find φ, you need to know the initial conditions: the initial position x(0) and the initial velocity v(0).
Step 4: Substitute t = 0 into the equation to find the initial position: x(0) = A cos(φ).
Step 5: If you know x(0), you can rearrange the equation to solve for φ: φ = cos⁻¹(x(0)/A).
Step 6: To find the phase constant more accurately, also consider the initial velocity: v(0) = -Aω sin(φ).
Step 7: Use both x(0) and v(0) to determine φ, ensuring that the values are consistent with the motion.
