A 326 g object is attached to a spring and oscillates with a period of 0.25 s. The total energy in this system is 5.83 J. What is the amplitude of the motion?

The total energy of the system is the spring potential energy at maximum displacement, which is the Amplitude, X

(1/2) k X^2 = 5.83 J

You need the spring constant, k, to compute X. Get it from the period, P = 0.25 s

P = 2 pi sqrt(M/k) = 0.25 s

sqrt(M/k) = 0.25/(2 pi) = 0.0398 s
M/k = 1.583*10^-3 s^2
k = 205.9 N/m

Now solve the first equation for X

To find the amplitude of the motion, we need to use the formula for the total energy in a simple harmonic motion system.

The formula for the total energy in a simple harmonic motion system is given by:

Total Energy = (1/2) * k * A^2

Where:
- Total Energy is the total energy in the system
- k is the spring constant
- A is the amplitude of the motion

We have been given the total energy (5.83 J), and we need to find the amplitude (A).

First, let's rearrange the formula to solve for A:

Total Energy = (1/2) * k * A^2

Multiply both sides of the equation by 2:

2 * Total Energy = k * A^2

Now, divide both sides of the equation by k:

(2 * Total Energy) / k = A^2

Finally, take the square root of both sides of the equation to solve for A:

A = sqrt((2 * Total Energy) / k)

To calculate the amplitude, we still need the spring constant (k). The period of the oscillation is given as 0.25 s. The period (T) of an object undergoing simple harmonic motion is related to the spring constant (k) and the mass (m) of the object by the formula:

T = 2π * sqrt(m / k)

Given the period (0.25 s), we can rearrange the formula to solve for the spring constant (k):

k = (4 * π^2 * m) / T^2

The mass of the object is given as 326 g, which we need to convert to kilograms (kg) before using it in the formula:

mass (m) = 326 g = 0.326 kg

Substituting the values into the formula, we can calculate the spring constant (k):

k = (4 * π^2 * 0.326) / (0.25^2)

Now that we have the value of k, we can substitute it into the formula we derived earlier to find the amplitude (A):

A = sqrt((2 * Total Energy) / k)

Substituting the values of the total energy (5.83 J) and the spring constant (k), we can calculate the amplitude (A).