When is a Pendulum in Simple Harmonic Motion?(SHM)

Summary

In this educational video, the speaker discusses the principles of simple harmonic motion (SHM) using a mass-spring system and a pendulum. The key features of SHM, such as restoring force, displacement, velocity, and acceleration, are explored and compared between the two systems. The video explains how the pendulum's restoring force is the tangential component of gravity, which is proportional to angular displacement, and how small angle approximations (up to 15 degrees) allow the pendulum to exhibit SHM. This engaging explanation clarifies the connection between pendulums and SHM, with an emphasis on the conditions for accurate motion modeling.

Takeaways

• 😀 A restoring force is required for an object to be in simple harmonic motion (SHM).
• 😀 The restoring force must be proportional to the displacement from the equilibrium position and directed toward the equilibrium.
• 😀 In a mass-spring system, displacement is linear, while in a pendulum, displacement is angular (measured in degrees or radians).
• 😀 The symbol for angular displacement in a pendulum's SHM is theta (θ), and it is typically measured in radians for trigonometric calculations.
• 😀 At extreme positions (Positions 1 and 3), the velocity of an object in SHM is zero, while at the equilibrium position (Position 2), velocity is maximum.
• 😀 The acceleration in SHM is maximum at extreme positions (Positions 1 and 3) and zero at the equilibrium position (Position 2).
• 😀 The restoring force for a pendulum in SHM is the component of the force of gravity that acts tangential to the motion.
• 😀 The force of gravity itself is not the restoring force, as it is constant and not proportional to the displacement from the equilibrium position.
• 😀 The force of tension in the string of the pendulum is not the restoring force, as it is greatest at the equilibrium position and not proportional to displacement.
• 😀 A pendulum is considered to be in SHM for small amplitudes, typically less than 15 degrees, due to the small angle approximation, where sine of the angle is approximately equal to the angle itself in radians.

Q & A

What are the main conditions required for an object to be in simple harmonic motion?

For an object to be in simple harmonic motion, there must be a restoring force that is always directed toward the rest position, and this force must be proportional to the displacement from the rest position.

What is the difference between the displacement in a mass-spring system and a pendulum?

In a mass-spring system, the displacement is linear, whereas in a pendulum, the displacement is angular. For the pendulum, we use angular displacement, denoted by the symbol theta, instead of linear displacement (x).

Why is the displacement in a pendulum described as angular rather than linear?

The displacement in a pendulum is described as angular because the pendulum moves in an arc along a circular path, meaning the displacement is measured as an angle (theta), unlike linear displacement in a mass-spring system.

What is the significance of using radians when working with the equations for simple harmonic motion in a pendulum?

When working with the trigonometric equations for simple harmonic motion in a pendulum, the angular displacement (theta) must be in radians because the mathematical relationships (like sine) are based on the radian measure.

What happens to the velocity of an object in simple harmonic motion at extreme positions (Positions 1 and 3)?

At the extreme positions (Positions 1 and 3), where the displacement is at its maximum (amplitude), the velocity of the object is zero because the object is momentarily stationary before changing direction.

How does the acceleration of an object in simple harmonic motion behave at the extreme positions and at the equilibrium position?

At the extreme positions (Positions 1 and 3), the displacement is maximum, so the restoring force and acceleration are also maximum. At the equilibrium position (Position 2), the displacement is zero, so the restoring force and acceleration are zero.

What forces are acting on a pendulum, and which one is responsible for restoring it to its equilibrium position?

The two forces acting on a pendulum are the force of gravity and the force of tension in the string. The restoring force is the component of gravity that acts in the tangential direction to the motion of the pendulum.

Why can't the force of tension be the restoring force in a pendulum?

The force of tension cannot be the restoring force because its magnitude is greatest when the pendulum is at the equilibrium position, not when the pendulum is displaced. The restoring force needs to be proportional to the displacement from the equilibrium position.

How does the force of gravity contribute to the restoring force in a pendulum's simple harmonic motion?

The restoring force in a pendulum is due to the tangential component of the force of gravity. This component increases as the angular displacement increases, and it is directed toward the equilibrium position, satisfying the conditions for simple harmonic motion.

What is the small angle approximation, and how does it relate to a pendulum's simple harmonic motion?

The small angle approximation states that for small angles (typically less than 15 degrees), the sine of the angle is approximately equal to the angle itself when measured in radians. This approximation allows a pendulum to exhibit simple harmonic motion for small amplitudes, typically considered to be less than 15 degrees.