To determine the maximum speed of a mass attached to a spring that has been displaced and released, we can use the formula for the maximum speed in simple harmonic motion:
[tex]\[ v_{\text{max}} = \sqrt{\frac{k}{m}} \cdot A \][/tex]
where:
- [tex]\( v_{\text{max}} \)[/tex] is the maximum speed,
- [tex]\( k \)[/tex] is the spring constant,
- [tex]\( m \)[/tex] is the mass,
- [tex]\( A \)[/tex] is the amplitude of the motion (which in this case is the initial displacement).
Given the data:
- Spring constant, [tex]\( k = 33.5 \, \text{N/m} \)[/tex]
- Mass, [tex]\( m = 1.20 \, \text{kg} \)[/tex]
- Displacement (amplitude), [tex]\( A = 0.120 \, \text{m} \)[/tex]
Let's go through the steps to calculate the maximum speed:
1. First, determine the ratio of the spring constant to the mass:
[tex]\[
\frac{k}{m} = \frac{33.5 \, \text{N/m}}{1.20 \, \text{kg}}
\][/tex]
2. After calculating the ratio, take the square root of this ratio to get the angular frequency:
[tex]\[
\sqrt{\frac{k}{m}}
\][/tex]
3. Multiply this result by the amplitude [tex]\( A \)[/tex]:
[tex]\[
v_{\text{max}} = \sqrt{\frac{33.5}{1.20}} \cdot 0.120
\][/tex]
Following these steps and using the correct values, the maximum speed comes out to be:
[tex]\[ v_{\text{max}} \approx 0.634 \, \text{m/s} \][/tex]
So, the maximum speed of the mass is approximately [tex]\( 0.634 \, \text{m/s} \)[/tex].
