The period of a simple pendulum (7) is measured as a function of length (L) on an unknown planet. The dependence of of L is linear (see the graph below). Use the slope of this graph to calculate the acceleration of gravity on this planet, in m/s2. 0.25 y = 0.0909x 0.2 0.15 0.05 Plot Area 0.5 1 1.5 2.5 L (m) T2/4 pi? (s²)
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.
![**Understanding Pendulum Motion on an Unknown Planet**
*Topic: Calculating Gravitational Acceleration*
The period of a simple pendulum (\( T \)) is measured as a function of length (\( L \)) on an unknown planet. The dependence of \( \frac{T^2}{4\pi^2} \) on \( L \) is linear, as seen in the provided graph. Use the slope of this graph to calculate the acceleration of gravity on this planet, in \( \text{m/s}^2 \).
**Graph Explanation**
- **X-Axis**: Represents the length of the pendulum (\( L \)) in meters (m), ranging from 0 to 3 meters.
- **Y-Axis**: Represents \( \frac{T^2}{4\pi^2} \) in seconds squared (\( \text{s}^2 \)), ranging from 0 to 0.25 \( \text{s}^2 \).
- **Data Points**: Various plotted points showing the relationship between \( L \) and \( \frac{T^2}{4\pi^2} \).
**Fitting Line**: There is a linear trend with the equation of the line given by:
\[ y = 0.0909x \]
In this context:
- \( y \) represents \( \frac{T^2}{4\pi^2} \)
- \( x \) represents the pendulum length \( L \)
**Calculating Gravitational Acceleration**
The equation for the period \( T \) of a simple pendulum is:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
Squaring both sides and rearranging to match the linear form \( \frac{T^2}{4\pi^2} = \frac{L}{g} \), we can see that the slope of this line (\( \frac{T^2}{4\pi^2} \) vs. \( L \)) is \( \frac{1}{g} \).
Given the slope from the graph:
\[ \text{Slope} = 0.0909 = \frac{1}{g} \]
Thus, the acceleration due to gravity \( g \) on this unknown planet can be calculated as follows:
\[ g = \frac{1}{0.0909} \approx 11 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00e5c2ed-ec3e-4354-a60a-b0d8fa7c3143%2Ffacf2b8a-f657-4fcb-b87b-5a9f9d9358dd%2Flt97hm_processed.png&w=3840&q=75)
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