2. An object weighing 44.1 N hangs from a vertical massless ideal spring. When set in vertical motion, the object obeys the equation y(t) = (6.20 cm) cos[(2.74 rad/s)t - 1.40]. a. Find the time for this object to vibrate one complete cycle. ( b. What are the maximum speed and maximum acceleration of the object? (
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.
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![**Physics Problem Involving Harmonic Motion**
**Problem Statement:**
An object weighing 44.1 N hangs from a vertical massless ideal spring. When set in vertical motion, the object obeys the equation \( y(t) = (6.20 \text{ cm}) \cos[(2.74 \text{ rad/s}) t - 1.40] \).
**Questions:**
a. Find the time for this object to vibrate one complete cycle.
b. What are the maximum speed and maximum acceleration of the object?
**Solution:**
a. To find the time for the object to vibrate one complete cycle, we need to determine the period \( T \) of the oscillation. The period \( T \) is related to the angular frequency \( \omega \) by the following formula:
\[ T = \frac{2\pi}{\omega} \]
Given \( \omega = 2.74 \text{ rad/s} \),
\[ T = \frac{2\pi}{2.74 \text{ rad/s}} \approx \frac{6.2832}{2.74} \approx 2.29 \text{ seconds} \]
b. The maximum speed \( v_{\text{max}} \) and maximum acceleration \( a_{\text{max}} \) can be determined from the following relationships:
- Maximum speed:
\[ v_{\text{max}} = \omega A \]
where \( A \) is the amplitude of the motion.
Given \( A = 6.20 \text{ cm} = 0.062 \text{ m} \) and \( \omega = 2.74 \text{ rad/s} \),
\[ v_{\text{max}} = 2.74 \text{ rad/s} \times 0.062 \text{ m} \approx 0.170 \text{ m/s} \]
- Maximum acceleration:
\[ a_{\text{max}} = \omega^2 A \]
Given \( A = 0.062 \text{ m} \) and \( \omega = 2.74 \text{ rad/s} \),
\[ a_{\text{max}} = (2.74 \text{ rad/s})^2 \times 0.062 \text{ m} \approx 0.466 \text{ m/s}^2 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc88a4be5-ce0c-4e25-871d-a0bd16127b90%2F60a120d1-99c3-4e9b-a671-e2edd474ef03%2F5vw0nd5_processed.jpeg&w=3840&q=75)
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