A ball is attached to a spring on a frictionless horizontal surface. The ball is pulled to the right and released from rest at t = 0 s. If the ball reaches a maximum speed of 34.1 cm/s and oscillates with a period of 4.50 s, what is the ball's velocity at t = 0.350 s?
To find the ball's velocity at t = 0.350 s, we can use the concept of simple harmonic motion (SHM) along with the given information.
In SHM, the movement of an object is characterized by a sinusoidal function. The general equation for displacement in SHM is given by:
x(t) = A * cos(ωt + φ)
Where:
- x(t) represents the displacement of the ball at time t.
- A is the amplitude of the motion.
- ω is the angular frequency of the motion.
- φ is the phase constant.
The period of oscillation (T) is related to the angular frequency by the formula:
T = 2π/ω
Given that the period is 4.50 s, we can calculate the angular frequency:
ω = 2π/T
= 2π/4.50 s
= 1.396 rad/s
Now, let's analyze the given information about the ball's maximum speed and the oscillation period.
At the maximum speed (v_max), the ball is at its equilibrium position, where the displacement (x) is zero. Therefore, we have:
x(t) = A * cos(φ) = 0
Since the ball is pulled to the right and released from rest, it will start in the positive direction. This means the initial phase constant (φ) is zero.
Now, let's determine the amplitude (A) from the given maximum speed. At the maximum speed, the ball's velocity (v) is equal to the angular frequency (ω) multiplied by the amplitude (A):
v_max = |A * ω|
Substituting the given values, we can solve for A:
34.1 cm/s = |A * 1.396 rad/s|
A = 24.43 cm
Therefore, the amplitude of the motion is 24.43 cm.
Now, we have all the necessary parameters to determine the displacement equation:
x(t) = 24.43 cm * cos(1.396t)
Finally, to find the ball's velocity at t = 0.350 s, we need to differentiate the displacement equation with respect to time:
v(t) = dx(t)/dt = -24.43 cm * sin(1.396t) * (1.396 rad/s)
Now, substitute t = 0.350 s into the equation to get the velocity at that particular time:
v(0.350 s) = -24.43 cm * sin(1.396 * 0.350) * (1.396 rad/s)
Calculating this expression will give you the velocity of the ball at t = 0.350 s.