L-10 Uniform Circular Motion: Kinematics and Dynamics

Uniform Circular Motion

Kinematics of Uniform Circular Motion

Uniform circular motion refers to the motion of an object traveling in a circle with constant speed. Although the speed remains constant, the direction of the velocity changes continuously, resulting in acceleration.

• Definition: Uniform circular motion is the motion of an object in a circle of fixed radius at constant speed.

• Velocity Direction: The instantaneous velocity vector is always tangent to the circle at any point.

• Acceleration: The change in velocity over time, even at constant speed, leads to a nonzero acceleration directed toward the center of the circle.

• Centripetal Acceleration Formula:

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

• Direction: Centripetal acceleration always points toward the center of the circle.

• Example: A car moving at constant speed around a circular track experiences a change in direction of velocity, resulting in centripetal acceleration.

Dynamics of Uniform Circular Motion

For an object to maintain uniform circular motion, a net force must act on it, directed toward the center of the circle. This force is called the centripetal force.

• Net Force Requirement: The net force required for circular motion is given by:

$\sum F_R = m a_R = m \frac{v^2}{r}$

• Centripetal Force: The force that keeps an object moving in a circle, always directed toward the center.

• Centrifugal Force: In the reference frame of the rotating object, a fictitious outward force (centrifugal force) may be perceived, but it does not act in an inertial frame.

• Loss of Centripetal Force: If the centripetal force vanishes, the object moves off in a straight line tangent to the circle.

• Example: When a ball on a string is released, it flies off tangentially to the circle.

Applications and Examples

Example: Revolving Ball (Horizontal Circle)

Consider a ball of mass attached to a string, revolving in a horizontal circle. The tension in the string provides the necessary centripetal force.

• Key Points:

  • The tension in the string must equal the required centripetal force.

  • Ignoring the mass of the string and gravity (if the circle is perfectly horizontal), the force is:

$F_T = m \frac{v^2}{r}$

• Example: A 0.150 kg ball revolves in a horizontal circle of radius 0.600 m at a certain speed. Calculate the tension required.

Example: Revolving Ball (Vertical Circle)

When a ball is swung in a vertical circle, both gravity and tension contribute to the net force at different points in the motion.

• Minimum Speed at Top: To maintain circular motion at the top of the arc, the minimum speed is found by setting the tension to zero (just before the string would go slack):

$m \frac{v^2}{r} = mg \implies v_{min} = \sqrt{g r}$

• Tension at Bottom: At the bottom, both gravity and centripetal force act upward, so:

$T_{bottom} = m \frac{v^2}{r} + mg$

• Example: Calculate the minimum speed and tension for a ball on a 1.10 m cord.

Physics Applied: Highway Curves

Forces on a Car Rounding a Curve

When a car rounds a curve, a net force toward the center of the curve is required to change the direction of the car's velocity. On a flat road, this force is provided by friction between the tires and the road.

• Static Friction: If the tires do not slip, the friction is static and can point toward the center of the circle.

• Kinetic Friction: If the tires slip, friction becomes kinetic, which is less than static friction and points opposite to the direction of motion, making it difficult to regain control.

• Maximum Speed Without Skidding: The maximum speed at which a car can round a curve without skidding is determined by the coefficient of static friction:

$v_{max} = \sqrt{\mu_s g r}$

• Example: For a curve of radius 125 m and coefficient of static friction $\mu_s = 0.60$, calculate the maximum safe speed.

Example: Skidding on a Curve

Given a car rounding a curve at 15 m/s, determine if it will skid under different road conditions.

• Dry Pavement: $\mu_s = 0.60$

• Icy Pavement: $\mu_s = 0.25$

• Compare the required centripetal force to the maximum static friction force to determine if skidding occurs.

Summary Table: Forces in Circular Motion

Situation	Net Force Toward Center	Formula
Horizontal Circle (string tension)	Tension in string	$F_T = m \frac{v^2}{r}$
Vertical Circle (top)	Tension + gravity	$T_{top} + mg = m \frac{v^2}{r}$
Vertical Circle (bottom)	Tension - gravity	$T_{bottom} - mg = m \frac{v^2}{r}$
Car on flat curve	Static friction	$f_s = m \frac{v^2}{r}$

Additional info: Table entries inferred from standard physics curriculum for circular motion.