An air-track cart is attached to a spring and completes one oscillation every 5.67 s in simple harmonic motion. At time t=0.00 s the cart is released at the position x=+0.250 m. What is the position of the cart when t=29.6 s ?
A) x=0.218 m
B) x=0.0461 m
C) x=0.210 m
D) x=0.342 m
E) x=-0.218 m

Question

An air-track cart is attached to a spring and completes one oscillation every  5.67 s in simple harmonic motion. At time  t=0.00 s the cart is released at the position  x=+0.250 m. What is the position of the cart when  t=29.6 s ?
A)  x=0.218 m
B)  x=0.0461 m
C)  x=0.210 m
D)  x=0.342 m
E)  x=-0.218 m

Quizplus · Accepted Answer

The position of the cart in simple harmonic motion can be determined using the formula $x = A \cos(2\pi \frac{t}{T} + \phi)$, where $A$ is the amplitude, $T$ is the period, $t$ is the time, and $\phi$ is the phase constant. Given that the cart completes one oscillation every 5.67 s, the period $T = 5.67$ s. The cart is released at $x = +0.250$ m at $t = 0.00$ s, which means $A = 0.250$ m and $\phi = 0$ (since it starts at the maximum displacement). To find the position at $t = 29.6$ s, we substitute these values into the formula:$$x = 0.250 \cos\left(2\pi \frac{29.6}{5.67}\right)$$The fraction $\frac{29.6}{5.67}$ gives the number of cycles completed, and since we're interested in the position at a specific time, we need to calculate the cosine of this fraction times $2\pi$. The exact value will depend on the calculator, but the key is recognizing that the position repeats every period, and 29.6 s is equivalent to some whole number of periods plus a fraction of a period. Calculating the exact position requires a calculator for the cosine value, but the principle is that the position is a function of the cosine of the time divided by the period, multiplied by $2\pi$, which gives us the phase of the motion at that time. Since the question is about the qualitative understanding and calculation of the position, the exact numerical calculation would show that the position is closest to the value given in option B, $x = 0.0461$ m, based on the periodic nature of the cosine function and the given parameters.

An Air-Track Cart Is Attached to a Spring and Completes