Uniform Circular Motion and Centripetal Forces: Study Notes

Uniform Circular Motion (UCM)

Definition and Key Concepts

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 (UCM): Motion of an object 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 in UCM

The period ($T$) is the time it takes to complete one revolution. 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

Even though the speed is constant in UCM, the velocity is not, because its direction changes. This change in direction means the object is accelerating toward the center of the circle. This acceleration is called centripetal acceleration ($a_c$).

• Direction: Always points inward, toward the center of the circular path.

• Formula for centripetal acceleration:

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

• 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 ($F_c$), which always points toward the center of the circle.

• Formula for centripetal force:

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

• 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) and Centripetal Force

When analyzing circular motion, the centripetal force should not be shown as a separate force in a free body diagram. Instead, it is the vector sum of all radial forces.

• Example: If several forces act on an object in circular motion, the centripetal force is the sum of those pointing toward the center minus those pointing away.

$F_c = F_1 + F_2 - F_3$

Applications of Centripetal Force

Car on a Curve

When a car moves along a curved path, the frictional force between the tires and the road provides the necessary centripetal force.

• Maximum safe speed: Occurs when the maximum static friction equals the required centripetal force.

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

Solving for $v$:

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

• Example: For $r = 50$ m, $\mu_s = 0.9$, $g = 9.8$ m/s$^2$:

$v = \sqrt{50 \times 0.9 \times 9.8} = 21$ 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 provides the centripetal force. The angle can be found by resolving the forces into components.

• Vertical equilibrium: $T_y = W$

• Horizontal (centripetal): $T_x = \frac{m v^2}{r}$

• Angle formula:

$\tan \theta = \frac{T_x}{T_y} = \frac{g r}{v^2}$

• Example: For $r = 1.5$ m, $v = 2.7$ m/s, $g = 9.8$ m/s$^2$:

$\tan \theta = \frac{9.8 \times 1.5}{(2.7)^2} = 2.0 \Rightarrow \theta = 64^\circ$

Satellites in Circular Orbits

Projectile Motion and Orbits

Satellites orbit the Earth by moving fast enough horizontally that as they fall due to gravity, the Earth's surface curves away beneath them. This results in continuous free fall around the planet.

• Only force acting: Gravitational force, which provides the centripetal force for the orbit.

• Formula for gravitational force:

$F_g = \frac{G M m}{r^2}$

• Setting gravitational force equal to centripetal force:

$\frac{G M m}{r^2} = \frac{m v^2}{r}$

Solving for orbital speed:

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

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

• Radius for geosynchronous orbit:

$r = \left( \frac{G M T^2}{4 \pi^2} \right)^{1/3}$

• Altitude above Earth's surface: Subtract Earth's radius from $r$.

Weightlessness and Artificial Gravity

Weightlessness in Orbit

Astronauts in orbit experience apparent weightlessness because both they and their spacecraft are in continuous free fall toward Earth. The sensation is similar to being in a falling elevator.

• True weightlessness: Occurs when gravity is the only force acting, and all objects accelerate at the same rate.

• Physiological effects: Extended weightlessness can affect the human body.

Artificial Gravity

Artificial gravity can be created in space by rotating a space station, causing a centripetal force that mimics the sensation of gravity.

• Einstein's Principle of Equivalence: A uniformly accelerated reference frame is equivalent to a gravitational field in the opposite direction.

• Application: Rotating space stations can simulate gravity for comfortable living.

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 in the circle.

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

• Minimum speed at the top: Required to maintain contact with the track.

$N + W = \frac{m v^2}{r}$

For minimum speed, $N = 0$:

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

Weightlessness in Parabolic Flight

During parabolic flight (e.g., NASA's C-9 aircraft), passengers experience brief periods of weightlessness as the plane follows a trajectory that approximates circular motion at the top of the arc.

• Condition for weightlessness: The normal force ($F_N$) becomes zero.

Summary Table: Key Formulas in Circular Motion

Quantity	Formula	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 inward force
Orbital Speed ($v$)	$v = \sqrt{\frac{G M}{r}}$	Speed for circular orbit
Minimum Speed at Top of Loop ($v_{min}$)	$v_{min} = \sqrt{g r}$	Required to stay on track