Air-Track Cart Oscillating on a Spring, solve for time

In summary, an air-track cart oscillating on a spring is a scientific demonstration used to study energy and motion. The cart is attached to a spring and when released, it oscillates back and forth. To solve for time, one can use equations involving the period, mass, spring constant, length, and acceleration due to gravity. Factors such as mass, spring stiffness, length, and gravity can affect the time of oscillation. The amplitude of oscillation does not affect the time, but it does impact the energy of the system. This demonstration can help us understand the relationship between potential and kinetic energy, conservation of energy, and periodic motion.
  • #1
Becca93
84
1
Homework Statement
The position of a an air-track cart that is oscillating on a spring is given by (14.0cm)cos[(12.0s-1)t]. At what value t after t=0 is the cart first located at x=2.0 cm?


The attempt at a solution

y = Acos(wt+∅)

y = 14cos(12t)
2 = 14cos(12t)
(1/7) = cos(12t)
12t = cos^-1(1/7)
t = cos^/1(1/7) / 12
t = 6.816 seconds,
incorrect.

Does anyone have any advice on how to proceed with this question?
 
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  • #2
When finding the angle, be sure to express it in radians. cos^-1(1/7) = ??
 
  • #3
Doc Al said:
When finding the angle, be sure to express it in radians. cos^-1(1/7) = ??

Thank you!
 

1. What is an air-track cart oscillating on a spring?

An air-track cart oscillating on a spring is a simple scientific demonstration used to study the properties of energy and motion. It consists of a cart that is free to move on a low-friction air track and is attached to a spring. When the cart is pulled and released, it will oscillate back and forth due to the energy stored in the spring.

2. How do you solve for time in an air-track cart oscillating on a spring?

To solve for time in an air-track cart oscillating on a spring, you can use the equation T=2π√(m/k), where T is the period of oscillation, m is the mass of the cart, and k is the spring constant. You can also use the equation T=2π√(L/g), where L is the length of the spring and g is the acceleration due to gravity. Both equations will give you the same result for the period of oscillation.

3. What factors can affect the time of oscillation in an air-track cart oscillating on a spring?

The time of oscillation in an air-track cart oscillating on a spring can be affected by several factors. These include the mass of the cart, the stiffness of the spring (measured by the spring constant), the length of the spring, and the acceleration due to gravity. Changes in any of these factors can alter the period of oscillation and affect the time it takes for the cart to complete one full cycle of oscillation.

4. How does the amplitude of oscillation affect the time of oscillation in an air-track cart?

The amplitude of oscillation, or the maximum displacement of the cart from its equilibrium position, does not affect the time of oscillation in an air-track cart. The period of oscillation is only dependent on the mass, spring stiffness, and length of the spring, not the amplitude. However, the amplitude can affect the energy of the system, as a larger amplitude will result in a larger maximum potential and kinetic energy.

5. What can an air-track cart oscillating on a spring teach us about energy and motion?

An air-track cart oscillating on a spring can teach us about the relationship between potential and kinetic energy, as well as the conservation of energy. The cart's motion is a result of the energy stored in the spring, which is converted between potential and kinetic energy as it oscillates. This demonstration can also help us understand the concept of periodic motion and how different factors can affect the period of oscillation.

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