A 50-g mass is attached to a spring and undergoes simple harmonic motion. Its maximum acceleration is 15m/s2 and its maximum speed is 3.5m/s .
Determine the angular frequency.
Determine the spring constant.
Determine the amplitude of the motion.
1) 4.3 rad/s
2) 2.7 N/m
3) 0.81 m
To determine the angular frequency, spring constant, and amplitude, we can use the formulas and principles of simple harmonic motion.
1. To find the angular frequency (ω):
The maximum acceleration (a_max) of an object in simple harmonic motion is given by the formula: a_max = ω^2 * amplitude
Rearranging the formula, we get: ω = sqrt(a_max / amplitude)
Substituting the given values, where a_max = 15 m/s², we can rewrite the formula as:
ω = sqrt(15 / amplitude)
2. To find the spring constant (k):
The formula for the angular frequency (ω) of an object undergoing simple harmonic motion is: ω = sqrt(k / m)
Rearranging the formula, we get: k = ω^2 * m
Substituting the given values, where m = 50 g = 0.05 kg and ω is calculated from step 1, we can rewrite the formula as:
k = ω^2 * m
3. To find the amplitude:
The maximum speed (v_max) of an object in simple harmonic motion is given by the formula: v_max = ω * amplitude
Rearranging the formula, we get: amplitude = v_max / ω
Substituting the given values, where v_max = 3.5 m/s and ω is calculated from step 1, we can rewrite the formula as:
amplitude = 3.5 / ω
Now we can calculate the values step by step:
Step 1:
ω = sqrt(15 / amplitude)
Step 2:
k = ω^2 * m
Step 3:
amplitude = 3.5 / ω
By following these steps, you should be able to determine the angular frequency, spring constant, and amplitude of the simple harmonic motion.