Circular Motion
Circular motion refers to the movement of a particle or object along a circular path. It can occur in two dimensions (e.g., a car on a circular track) or in three dimensions (e.g., the orbit of a planet around the Sun).
Uniform and Non-Uniform Circular Motion
- Uniform Circular Motion (UCM):
- In UCM, the object moves with a constant speed along the circular path.
- Although the speed is constant, the velocity changes because its direction changes continuously.
- Example: A satellite orbiting Earth at a constant altitude.
- Non-Uniform Circular Motion:
- In non-uniform circular motion, the speed of the object varies as it moves along the circular path.
- Both the magnitude and direction of velocity change.
- Example: A car accelerating along a curved highway.
Important Terms in Circular Motion
1. Time Period ([math]T[/math])
- The time period is the time taken by the object to complete one full revolution along the circular path.
- Formula: [math]T = \frac{\text{Total Time Taken}}{\text{Number of Revolutions}}[/math]
- Units: Seconds (s)
2. Frequency ([math]f[/math])
- The frequency is the number of revolutions completed by the object per unit time.
- Relation with Time Period: [math]f = \frac{1}{T}[/math]
- Units: Hertz (Hz)
3. Angular Displacement ([math]\Delta \theta[/math])
- The angular displacement is the angle subtended by the radius vector of the circular path at the center during the motion.
- It is measured in radians and is given as: [math]\Delta \theta = \frac{\text{Arc Length}}{\text{Radius}}[/math]
- Units: Radians (rad)
4. Angular Frequency ([math]\omega[/math])
- The angular frequency is the rate at which the angular displacement changes with time. It represents how fast the object is moving around the circle in terms of angle.
- Relation with Time Period: [math]\omega = \frac{2\pi}{T}[/math]
- Relation with Frequency: [math]\omega = 2\pi f[/math]
- Units: Radians per second (rad/s)
Centripetal Force
Centripetal force is the force that keeps a body moving in a circular path. It is always directed toward the center of the circle. Without this force, the object would move in a straight line due to inertia, as per Newton’s first law of motion.
Characteristics:
- Acts perpendicular to the velocity of the object.
- Changes only the direction of the velocity, not its magnitude.
- Examples:
- Gravitational force acts as the centripetal force for planetary orbits.
- Tension in a string provides the centripetal force for a pendulum.
Formula for Centripetal Force:
[math]F_c = \frac{m v^2}{r}[/math]
Where:
- [math]F_c[/math] = Centripetal force
- [math]m[/math] = Mass of the object
- [math]v[/math] = Speed of the object
- [math]r[/math] = Radius of the circular path
Centripetal Acceleration
Centripetal acceleration is the acceleration of an object moving in a circle at constant speed. It always points toward the center of the circle and is responsible for the change in the direction of the velocity vector.
Characteristics:
- It is perpendicular to the velocity of the object.
- It depends on the speed of the object and the radius of the circular path.
Formula for Centripetal Acceleration:
[math]a_c = \frac{v^2}{r}[/math]
Derivation of Centripetal Acceleration
Setup:
Consider an object moving in a circular path of radius [math]r[/math] with constant speed [math]v[/math]. Let the position vector of the object be [math]\vec{r_1}[/math] at point [math]P[/math] and [math]\vec{r_2}[/math] at point [math]Q[/math]. The velocity vectors at these points are [math]\vec{v_1}[/math] and [math]\vec{v_2}[/math], respectively.
Step 1: Velocity Change
The change in velocity vector is given by:
[math]\Delta \vec{v} = \vec{v_2} – \vec{v_1}[/math]
The angle between [math]\vec{r_1}[/math] and [math]\vec{r_2}[/math] is [math]\theta[/math]. Since the motion is uniform, the magnitude of velocity remains constant:
[math]|\vec{v_1}| = |\vec{v_2}| = v[/math]
Step 2: Similar Triangles
From the similarity of the triangles formed by the radius vectors and velocity vectors:
[math]\frac{\Delta v}{v} = \frac{\Delta r}{r}[/math]
Simplify:
[math]\Delta v = v \frac{\Delta r}{r}[/math]
Step 3: Acceleration
Acceleration is the rate of change of velocity:
[math]a_c = \frac{\Delta v}{\Delta t}[/math]
Substitute [math]\Delta v = v \frac{\Delta r}{r}[/math]:
[math]a_c = \frac{v}{r} \cdot \frac{\Delta r}{\Delta t}[/math]
Since [math]\frac{\Delta r}{\Delta t} = v[/math] (speed of the object):
[math]a_c = \frac{v^2}{r}[/math]
Final Summary
- Centripetal Force:
[math]F_c = \frac{m v^2}{r}[/math]
Centripetal Acceleration:
[math]a_c = \frac{v^2}{r}[/math]
Both the force and acceleration point toward the center of the circle, ensuring circular motion.
Non-Uniform Circular Motion
Non-uniform circular motion occurs when an object moves along a circular path with a changing speed. This motion involves two accelerations:
- Centripetal acceleration: Directed toward the center of the circle, responsible for changing the direction of velocity.
- Tangential acceleration: Acts along the tangent to the circle, responsible for changing the magnitude of velocity.
Tangential Acceleration
Tangential acceleration ([math]a_t[/math]) is the component of acceleration that causes a change in the speed (magnitude of velocity) of an object moving in a circular path.
Characteristics of Tangential Acceleration:
- Acts perpendicular to the radius of the circle.
- Affects the magnitude of velocity but does not influence its direction.
- Zero in uniform circular motion (where speed remains constant).
- It is responsible for increasing or decreasing the speed of the object.
Formula for Tangential Acceleration:
[math]a_t = \frac{d|v|}{dt}[/math]
Where:
- [math]a_t[/math] = Tangential acceleration
- [math]|v|[/math] = Speed of the object
- [math]t[/math] = Time
Net Acceleration in Non-Uniform Circular Motion
In non-uniform circular motion, the net acceleration ([math]\vec{a}[/math]) of the object is the vector sum of:
- Centripetal acceleration ([math]a_c[/math]):
- Directed toward the center of the circle.
- Magnitude: [math]a_c = \frac{v^2}{r}[/math].
- Tangential acceleration ([math]a_t[/math]):
- Directed along the tangent to the circle.
- Magnitude: [math]a_t = \frac{d|v|}{dt}[/math].
Formula for Net Acceleration:
Using the Pythagorean theorem, the net acceleration is:
[math]a = \sqrt{a_t^2 + a_c^2}[/math]
Direction of Net Acceleration
The direction of the net acceleration is determined by the resultant of the centripetal and tangential components. The angle [math]\theta[/math] between the net acceleration vector and the radius vector is given by:
[math]\tan\theta = \frac{a_t}{a_c}[/math]
Summary
- Tangential Acceleration:
[math]a_t = \frac{d|v|}{dt}[/math]
Centripetal Acceleration:
[math]a_c = \frac{v^2}{r}[/math]
Net Acceleration:
[math]a = \sqrt{a_t^2 + a_c^2}[/math]
Direction of Net Acceleration:
[math]\tan\theta = \frac{a_t}{a_c}[/math]
