Circular Motion and Gravitation: Concepts and Applications

Circular Motion and Gravitation

Introduction

This chapter explores the fundamental principles of circular motion and gravitation. It covers the kinematics and dynamics of objects moving in circles, the forces involved, and the universal law of gravitation. These concepts are essential for understanding planetary motion, satellite orbits, and many everyday phenomena.

Kinematics of Uniform Circular Motion

Definition and Key Concepts

• Uniform Circular Motion: Motion of an object in a circle at constant speed.

• Although speed is constant, the velocity (a vector) is always changing direction, so the object is accelerating.

• The acceleration responsible for this change in direction is called centripetal acceleration.

Centripetal Acceleration

• Always directed toward the center of the circle.

• Magnitude is given by: $a_c = \frac{v^2}{r}$ where $v$ is the speed and $r$ is the radius of the circle.

• Alternatively, using the period $T$ (time for one revolution): $a_c = \frac{4\pi^2 r}{T^2}$

Velocity in Circular Motion

• The instantaneous velocity is always tangent to the circle.

• Direction of velocity changes continuously, even if speed is constant.

Period and Frequency

• Period (T): Time taken for one complete revolution.

• Frequency (f): Number of revolutions per second ($f = 1/T$).

• Relationship between speed, radius, and period: $v = \frac{2\pi r}{T}$

Example: Acceleration of a Revolving Ball

• A ball of mass 0.150 kg is attached to a string and revolves in a horizontal circle of radius 0.600 m, making 2.00 revolutions per second.

• Find the centripetal acceleration: First, calculate the period: $T = \frac{1}{2.00} = 0.500$ s

• Speed: $v = \frac{2\pi r}{T} = \frac{2\pi \times 0.600}{0.500} = 7.54$ m/s

• Centripetal acceleration: $a_c = \frac{v^2}{r} = \frac{(7.54)^2}{0.600} = 94.7$ m/s$^2$

Summary Table: Key Quantities in Uniform Circular Motion

Quantity	Symbol	Formula	Units
Radius	r	–	m
Speed	v	$v = \frac{2\pi r}{T}$	m/s
Period	T	$T = \frac{1}{f}$	s
Frequency	f	$f = \frac{1}{T}$	Hz
Centripetal Acceleration	$a_c$	$a_c = \frac{v^2}{r}$	m/s$^2$

Applications

• Design of highways and racetracks (banked curves).

• Motion of planets and satellites.

• Everyday examples: spinning rides, rotating machinery.

Important Notes

• Centripetal means "center-seeking"; the force and acceleration always point toward the center of the circle.

• If the centripetal force is removed, the object moves off in a straight line tangent to the circle (Newton's first law).

Additional info:

• Further sections in the chapter (not shown in the images) likely cover the dynamics of circular motion, gravitational force, satellite motion, Kepler's laws, and types of forces in nature.