A cart connected to a spring oscillates according to the following position equation:
x = 0.20m cos (8 t)
What is the period of this oscillation?
What is the cart's velocity after 4 seconds?
Let P be the period
2 pi/P = 8
P = pi/4
velocity = dx/dt = -1.60 sin(8t) m/s
At t = 4 s,
V(8) = -1.60 sin64 = -1.47 m/s
(The 64 is radians, not degrees)
To find the period of the oscillation, we need to know the coefficient in front of the variable 't' in the argument of the cosine function. In this case, the coefficient is 8.
The formula for the period of an oscillation is given by T = (2π)/ω, where T is the period and ω is the angular frequency. The angular frequency, ω, is equal to 2π divided by the coefficient in front of 't'.
So, in this case, ω = 2π/8 = π/4.
Now, we can find the period:
T = (2π)/(π/4) = 8.
Therefore, the period of the oscillation is 8 seconds.
To find the cart's velocity after 4 seconds, we need to find the derivative of the position equation with respect to time (t).
Given x = 0.20m cos (8 t), we can take the derivative of x with respect to t:
v = dx/dt = -0.20m × 8 sin (8 t).
Substituting t = 4 seconds into this expression:
v = -0.20m × 8 sin (8 × 4) = -1.60 m/s.
Therefore, the cart's velocity after 4 seconds is -1.60 m/s (negative since the cart is moving in the opposite direction of positive x-axis).