The potential energy of an object attached to a spring is 2.70 J at a location where the kinetic energy is 1.20 J. If the amplitude A of the simple harmonic motion is 19.0 cm, calculate the spring constant k and the magnitude of the largest force Fypring, max that the object experiences. N k = m Fspring, max N II
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
![### Question 20 of 25
The potential energy of an object attached to a spring is 2.70 J at a location where the kinetic energy is 1.20 J. If the amplitude \( A \) of the simple harmonic motion is 19.0 cm, calculate the spring constant \( k \) and the magnitude of the largest force \( F_{\text{spring, max}} \) that the object experiences.
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\[ k = \quad \text{N} \hspace{-1mm}/\hspace{-1mm} \text{m} \]
\[ F_{\text{spring, max}} = \quad \text{N} \]
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In this problem, we need to determine two quantities related to a spring undergoing simple harmonic motion: the spring constant \( k \) and the maximum spring force \( F_{\text{spring, max}} \).
#### Given Data:
- Potential Energy \( U = 2.70 \) J
- Kinetic Energy \( K = 1.20 \) J
- Amplitude \( A = 19.0 \) cm (which is 0.190 m when converted to meters)
#### Relevant Equations:
1. The total mechanical energy in the system is the sum of the potential and kinetic energies:
\[
E_{\text{total}} = U + K
\]
2. The total mechanical energy can also be expressed in terms of the amplitude \( A \) and the spring constant \( k \):
\[
E_{\text{total}} = \frac{1}{2} k A^2
\]
3. The maximum force exerted by a spring \( F_{\text{spring, max}} \) can be found using Hooke's Law:
\[
F_{\text{spring, max}} = k A
\]
#### Steps to Solve:
1. Calculate the total mechanical energy:
\[
E_{\text{total}} = 2.70 \, \text{J} + 1.20 \, \text{J} = 3.90 \, \text{J}
\]
2. Use the total mechanical energy equation to solve for the spring constant \( k \):
\[
3.90 \, \text{J} = \frac{1}{2} k (0.190 \, \text{m})](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5039f9f1-9031-4cf0-b840-7355f91f2cd2%2F5461e6fc-4a1f-4d75-a0d3-103bd2ce1b66%2Fc3etxrq.png&w=3840&q=75)
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