We did a lab experiment about Hooke\'s law where we hung a suspended spring with

Question

We did a lab experiment about Hooke's law where we hung a suspended spring with some weights loaded on a hanger attached to the spring (we did the experiment twice with two different weight loads). I need help answering these three questions:

1. Newton's second law, -kx = Ma, contains the accelerated mass, M. Argue whether the mass of the hanger should be included (in computing the theoretical values, T', of the period).

2. Determine the percent error in the calculated value if T' (the theoretical values of the period) if the mass of the spring were neglected. Do this for both oscillating masses.

3. How would the period of vibrations be changed if the gravitational accleration were increased by 5%?

Please help! I will quickly give the points to the best answer.

Explanation / Answer

-kx = Ma

-kx = Md^2x/dt^2

Md^2x/dt^2 + kx = 0

solving

x = Asin(sqrt(M/k)t)

therefore Time period (T) = 2pi*sqrt(k/M)

1) As you can see from the expression of time period it depends upon the mass of the system, hence the total mass of the system should be considered to calculate the correct time period so you deifnitely need to include the mass of the hanger.

2) dT and dK stands for the error in T(time period) and K(spring constant) respectively.

    if you differentiate the the expression T= 2pi*sqrt(k/M) you will get

    dT = pi*(M^-1/2*k^-1/2*dK+k^-1/2*M^-3/2*dM)/MdT/T = 0.5dK/K + 0.5dM/M

    percentage error = dt/T*100 = 50*dM/M

    dM = mass of spring, M = total mass of system

3) As you can see the period of vibrations is independent of the gravitational acceleration hence increasing g will cause no effect on the time period of oscillation.

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