Set 9 Problem number 13

• Problem

• Solution

Problem

An ideal spring has an unknown restoring force constant. An mass of 6.874 kg suspended from the spring is observed to complete a cycle of oscillation in .69 seconds. What is the restoring force constant?

Solution

We know that the angular frequency of an object in simple harmonic motion is `omega = `sqrt(k/m).

The information given is that the object completes a cycle in .69 seconds.

• Thus the object completes 2 `pi ( 1.449) radians in .69 seconds.
• This implies a rate of

• angular frequency = 2 `pi radians/ ( .69 seconds) = 9.106 radians/second.

Since we know m, we know that 9.106 radians/second = `sqrt[k/( 2 kg)].

• Solving for k we obtain k = 6.874 kg( 9.106 radians/second) ^ 2 = 570 N/m.

The symbolic solution tell us that if `omega = `sqrt(k / m), then `omega^2 = k / m and m = k / `omega^2.

• Substituting the known k and `omega we obtain the same result as before.

In symbols, we solve `omega = `sqrt(k/m) for k, obtaining k = m * `omega ^ 2, then substitute the known values of m and the `omega found above.