# Pendulums | Oscillations and mechanical waves | Physics | Khan Academy

TLDRThis video script explores the concept of the simple pendulum as a type of simple harmonic oscillator. It explains that the restoring force and motion of a pendulum can be described by the simple harmonic oscillator equation, using the angular displacement instead of linear displacement. The script clarifies that while a simple pendulum is not a perfect harmonic oscillator, it closely resembles one, especially for small angles. The period of a pendulum is shown to be dependent on the length of the string and the gravitational acceleration, but not the mass of the pendulum bob. The formula for the period is provided, and it is emphasized that the approximation holds well for small angles, with deviations increasing as the angle approaches 70 degrees.

###### Takeaways

- π The script discusses simple harmonic oscillators, with a focus on pendulums as the second most common example after masses on springs.
- π A pendulum is a mass attached to a string of length L, which can be displaced and then oscillates back and forth like a simple harmonic oscillator.
- π The simple pendulum is considered simple because it consists of a single mass and string, unlike more complex systems like the double pendulum which can exhibit chaotic behavior.
- π The motion of a pendulum can be described using the simple harmonic oscillator equation, but with angular displacement (theta) instead of linear displacement (X).
- π The restoring force in a pendulum comes from gravity, which pulls the mass back towards its equilibrium position, creating an oscillating motion.
- π The period of a simple pendulum is not affected by the mass of the pendulum due to the canceling effects of increased inertia and gravitational force.
- π The period of a pendulum is directly proportional to the square root of the length of the string (L) and inversely proportional to the square root of the acceleration due to gravity (g).
- π The formula for the period of a pendulum is derived from calculus and involves the moment of inertia and torque, showing that increasing the string length increases the period.
- βοΈ Increasing the gravitational acceleration (g) decreases the period of the pendulum because a larger force results in greater acceleration and speed, completing the cycle faster.
- π The period formula for a pendulum, \( T = 2\pi \sqrt{\frac{L}{g}} \), is analogous to that of a mass on a spring, with the inertia term (mass or length) on top and the force term (spring constant or g) on the bottom.
- π« The simple pendulum's behavior as a simple harmonic oscillator and the accuracy of the period formula are only valid for small angular amplitudes, typically less than 20 degrees for minimal error.

###### Q & A

### What is the most common example of a simple harmonic oscillator?

-The most common example of a simple harmonic oscillator is a mass on a spring.

### What is the second most common example of a simple harmonic oscillator mentioned in the script?

-The second most common example is a pendulum, which is a mass, m, connected to a string of length, L, that swings back and forth.

### What is a simple pendulum and how does it differ from a more complicated pendulum system?

-A simple pendulum is a mass connected to a string without any additional masses or strings attached. A more complicated pendulum, such as a double pendulum, involves additional masses and strings, which can lead to chaotic motion and is much more complex to describe mathematically.

### Why do we study pendulums when studying simple harmonic oscillators?

-We study pendulums because they oscillate similarly to simple harmonic oscillators, and they can be described by the simple harmonic oscillator equation, especially when the angles are small.

### What variable is commonly used to describe the motion of a pendulum instead of displacement (X)?

-The variable commonly used to describe the motion of a pendulum is the angle (theta) that the pendulum makes with the vertical at any given time.

### What is the small angle approximation and why is it used for pendulums?

-The small angle approximation is an assumption that the pendulum's maximum angular displacement (amplitude) is small enough that the pendulum can be treated as a simple harmonic oscillator. It is used because for small angles, the deviation from true simple harmonic motion is minimal, typically less than 1% for angles less than 20 degrees.

### How does the mass of the pendulum affect its period of oscillation?

-Surprisingly, the mass of the pendulum does not affect its period of oscillation. This is because the increased inertia due to greater mass is offset by the increased gravitational force pulling the mass back towards equilibrium.

### What factors determine the period of a pendulum according to the formula provided in the script?

-The period of a pendulum is determined by the length of the string (L) and the acceleration due to gravity (g). The formula for the period is \( T = 2\pi \sqrt{\frac{L}{g}} \).

### Why does increasing the length of the string (L) increase the period of a pendulum?

-Increasing the length of the string increases the moment of inertia, which makes it harder to change the angular velocity of the pendulum. As a result, it takes longer to complete a cycle, thus increasing the period.

### Why does increasing the gravitational acceleration (g) decrease the period of a pendulum?

-Increasing the gravitational acceleration increases the force of gravity acting on the pendulum, which in turn increases the restoring force and the acceleration of the pendulum. This causes the pendulum to move faster and complete its cycle in less time, thus decreasing the period.

### How does the amplitude of the pendulum's swing affect the period, assuming small angles?

-For small angles, the amplitude of the pendulum's swing does not affect the period. This is because the pendulum can be approximated as a simple harmonic oscillator under these conditions.

### What is the error percentage of the period formula for pendulums when the maximum angle is less than 20 degrees?

-When the maximum angle is less than 20 degrees, the error percentage of the period formula for pendulums is less than 1%.

###### Outlines

##### π Introduction to Simple Harmonic Oscillators and Pendulums

The instructor begins by introducing simple harmonic oscillators, with masses on springs as the most common example, followed by pendulums. A pendulum is described as a mass attached to a string of length L that can swing back and forth. The video aims to explore the pendulum's behavior as a simple harmonic oscillator, with a focus on the simple pendulum for its uncomplicated setup. The instructor clarifies that while pendulums are technically not perfect simple harmonic oscillators, they are treated as such for small angular displacements due to their close approximation. The simple harmonic oscillator equation is introduced, with the variable X replaced by the angular displacement theta, to describe the pendulum's motion over time.

##### π The Effects of Mass on Pendulum Period

The instructor discusses the misconception that increasing the mass of a pendulum would affect its period. It is explained that while increasing mass increases inertia, making it harder to move, the gravitational force also increases, which could suggest a faster motion. However, these two effects cancel each other out, and the mass does not influence the period of the pendulum. The formula for the period of a simple pendulum is introduced without derivation, emphasizing that it requires calculus for a complete understanding. The instructor provides an intuitive explanation of why the variables in the formula, such as the length of the string (L) and the acceleration due to gravity (g), affect the period as they do.

##### β³ Factors Influencing the Period of a Simple Pendulum

The instructor explains that the period of a simple pendulum is influenced by the length of the string and the acceleration due to gravity. Increasing the length of the string increases the moment of inertia, making it harder to change the pendulum's angular velocity, thus increasing the period. Conversely, increasing the gravitational acceleration increases the restoring force, leading to a faster motion and a decreased period. The instructor draws an analogy between the pendulum's formula and that of a mass on a spring, highlighting the similarities in how inertia and force terms affect the period. It is also emphasized that the amplitude of the pendulum's swing does not affect the period, provided the angles are small, aligning with the small amplitude approximation for treating the pendulum as a simple harmonic oscillator.

###### Mindmap

###### Keywords

##### π‘Simple Harmonic Oscillator

##### π‘Pendulum

##### π‘Restoring Force

##### π‘Displacement

##### π‘Angular Displacement

##### π‘Amplitude

##### π‘Period

##### π‘Phase Constant

##### π‘Small Angle Approximation

##### π‘Moment of Inertia

##### π‘Torque

##### π‘Gravitational Acceleration

###### Highlights

A simple pendulum is the second most common example of a simple harmonic oscillator after masses on springs.

The simple pendulum consists of a mass m connected to a string of length L that swings back and forth.

The motion of a pendulum can be described by the simple harmonic oscillator equation with angular displacement.

The amplitude in the pendulum's equation is the maximum angular displacement, not the linear displacement.

A simple pendulum is technically not a perfect simple harmonic oscillator but is treated as such for small angles.

The period of a pendulum is not affected by the mass of the pendulum bob.

The period of a pendulum is directly proportional to the square root of the length of the string.

Increasing the length of the pendulum string results in an increased period of oscillation.

The moment of inertia of the pendulum is directly related to the square of the string's length.

The period of a pendulum is inversely proportional to the square root of the gravitational acceleration.

Increasing gravitational acceleration decreases the period of the pendulum due to increased restoring force.

The simple pendulum's period formula is analogous to that of a mass on a spring system.

Amplitude does not affect the period of a pendulum as long as the oscillations are within the small angle approximation.

The small angle approximation for a pendulum to act as a simple harmonic oscillator is typically less than 20 degrees.

For angles less than 20 degrees, the error in the pendulum's period formula is less than one percent.

The error in the pendulum's period formula is less than three percent for angles less than 40 degrees.

Even at angles up to 70 degrees, the error in the pendulum's period formula is less than ten percent.

The simple pendulum's period formula is derived from the principles of torque and moment of inertia, not requiring calculus for understanding.

###### Transcripts

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