Uniform Circular Motion and Centripetal Forces: Study Notes

Uniform Circular Motion (UCM)

Definition and Characteristics

Uniform Circular Motion (UCM) describes the motion of an object traveling at a constant speed along a circular path. Although the speed remains constant, the direction of the velocity changes continuously, resulting in acceleration.

• Uniform Circular Motion: Motion at constant speed along a circular path.

• Example: Spinning a ball on a string.

• Circumference of a circle: The distance for one revolution is $2\pi r$, where $r$ is the radius.

Period and Velocity

The period ($T$) is the time required to complete one full revolution around the circle. The relationship between period, radius, and velocity is given by:

• Period formula:

$T = \frac{2\pi r}{v}$

• Units: Seconds [s]

• Example calculation: For a plane traveling at 110 m/s around a circle of radius 2850 m:

$T = \frac{2\pi (2850\ \text{m})}{110\ \text{m/s}} = 163\ \text{s} \approx 2.72\ \text{min}$

Centripetal Acceleration

Direction and Calculation

Although the speed is constant in UCM, the velocity is not, because its direction changes. This change in direction means the object is accelerating, even if its speed does not change. The acceleration always points inward, toward the center of the circle, and is called centripetal acceleration ($a_c$).

• Centripetal acceleration: Acceleration directed toward the center of the circular path.

• Formula:

$a_c = \frac{v^2}{r}$

• Derivation: Based on the change in velocity over time as the object moves along the circle.

Centripetal Force

Definition and Application

According to Newton's Second Law, any acceleration must be caused by a net force. In UCM, this net force is called the centripetal force ($F_c$), and it always points toward the center of the circle.

• Centripetal force: The net force causing centripetal acceleration.

• Formula:

$F_c = m a_c = \frac{m v^2}{r}$

• Direction: Always toward the center of the circular path.

• Note: Centripetal force is not a new type of force; it is the sum of all radial forces acting toward the center.

Free Body Diagrams (FBD) in Circular Motion

Identifying Forces

When analyzing circular motion, it is important to identify all forces acting on the object. The centripetal force is the vector sum of these radial forces and should not be shown as a separate force in a free body diagram.

• Radial forces: Forces pointing toward or away from the center.

• Example: For a ball on a string, the tension ($T$) and weight ($W$) are the main forces.

• Fictitious forces: The sensation of being "thrown outward" is due to the so-called centrifugal force, which is not a real force but an effect of being in a non-inertial reference frame.

Solving Centripetal Force Problems

General Approach

To solve problems involving centripetal force, identify all radial forces and set their vector sum equal to the required centripetal force.

• Formula for vector sum:

$F_c = F_1 + F_2 - F_3$

• Example: For a car moving in a curve, the frictional force provides the centripetal force.

Force	Direction	Role in Circular Motion
Tension ($T$)	Toward center	Provides centripetal force for a ball on a string
Friction ($f_s$)	Toward center	Provides centripetal force for a car on a curve
Weight ($W$)	Downward	May contribute to or oppose centripetal force depending on orientation

Examples of Centripetal Force Applications

Car on a Curve

For a car moving at constant speed on an unbanked curve, the maximum safe speed is determined by the static friction between the tires and the road.

• Maximum speed formula:

$v = \sqrt{r \mu_s g}$

• Where: $r$ = radius of curve, $\mu_s$ = coefficient of static friction, $g$ = acceleration due to gravity.

• Example calculation: $v = \sqrt{(50)(0.9)(9.8)} = 21\ \text{m/s}$

Ball on a String at an Angle

When a ball is twirled on a string at an angle, the tension in the string has both vertical and horizontal components. The angle can be found using:

• Angle formula:

$\theta = \tan^{-1}\left(\frac{gr}{v^2}\right)$

• Example calculation: $\theta = \tan^{-1}\left(\frac{(9.8)(1.5)}{(2.7)^2}\right) = 64^\circ$

Satellites in Circular Orbits

Principles of Satellite Motion

Satellites orbit the Earth in circular paths due to the gravitational force acting as the centripetal force. The required orbital speed and radius are determined by the balance between gravity and the need for centripetal acceleration.

• Gravitational force provides centripetal force: $F_g = F_c$

• Orbital speed formula:

$v = \sqrt{\frac{G M}{r}}$

• Where: $G$ = gravitational constant, $M$ = mass of Earth, $r$ = radius of orbit.

• Geosynchronous satellites: Have an orbital period equal to Earth's rotation (24 hours).

Satellite Type	Orbital Period	Altitude Above Earth
Low Earth Orbit (LEO)	~90 min	~200-2,000 km
Geosynchronous Orbit	24 hr	~35,900 km

Weightlessness and Artificial Gravity

Weightlessness in Orbit

Astronauts in orbit experience apparent weightlessness because both they and their spacecraft are in free fall, accelerating toward Earth at the same rate. This is analogous to the sensation in a falling elevator.

• True weightlessness: Occurs when gravity is the only force acting and there is no contact force.

• Artificial gravity: Can be created by rotating a space station, using centripetal force to simulate gravity.

Vertical Circular Motion

Forces in Vertical Circles

When an object moves in a vertical circle (e.g., a motorcycle in a loop), the forces acting on it change depending on its position. The sum of the radial forces must always provide the required centripetal force.

• At the top of the loop: Both gravity and normal force point toward the center.

• Minimum speed at the top: To stay in contact with the track, the normal force must be at least zero.

$v_{min} = \sqrt{g r}$

• Where: $g$ = acceleration due to gravity, $r$ = radius of the loop.

Summary Table: Key Equations in Circular Motion

Quantity	Equation	Description
Period ($T$)	$T = \frac{2\pi r}{v}$	Time for one revolution
Centripetal Acceleration ($a_c$)	$a_c = \frac{v^2}{r}$	Acceleration toward center
Centripetal Force ($F_c$)	$F_c = \frac{m v^2}{r}$	Net force toward center
Orbital Speed ($v$)	$v = \sqrt{\frac{G M}{r}}$	Speed for satellite in orbit
Minimum Speed at Top of Loop ($v_{min}$)	$v_{min} = \sqrt{g r}$	Minimum speed to stay on track

Additional info: These notes expand on the original slides by providing full definitions, formulas, and context for each concept, as well as example calculations and tables for comparison.