PHYS 101 - LEC 9

Phys101 Lecture 9: Circular Motion

Key Points

• Centripetal Acceleration

• Dynamics of Uniform Circular Motion

Uniform Circular Motion—Kinematics

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

• Velocity: Instantaneous velocity is always tangent to the circle.

  • Magnitude of velocity (v): remains constant (v1 = v2 = v).

Centripetal Acceleration

• Definition: The acceleration directed towards the center of the circle during circular motion.

• Formula:[ a_R = \frac{v^2}{R} ]Where:

  • ( a_R ): Centripetal acceleration

  • ( v ): Tangential speed

  • ( R ): Radius of the circular path

Dynamics of Uniform Circular Motion

• Net Force Requirement: For an object to maintain uniform circular motion, it must have a net force acting towards the center, known as centripetal force.

• Centripetal Force Definition: The net force that keeps an object moving in a circular path, not to be counted as an additional force in free-body diagrams.

• Sources of Centripetal Force:

  • Tension in a string

  • Gravitational force acting on a satellite

  • Normal force in a ring or track

i-Clicker Question

• Question: An object is in uniform circular motion. Which statement must be true?

  • A. The net force acting on the object is zero.

  • B. The velocity of the object is constant.

  • C. The speed of the object is constant.

  • D. The acceleration of the object is constant.

Angular Velocity and Linear Velocity

• Angular Velocity: Defined in radians, measures how quickly an object rotates.

• Linear Velocity: Relates to angular velocity; as ( \Delta t ) approaches 0.

Example Problem

• Scenario: A ball of mass M hangs from a support via a massless string of length L, rotating to form an angle θ from vertical, making N revolutions per second.

• Objective: Derive an expression for the angle θ in terms of M, L, N, and necessary constants.

Dynamics of Circular Motion

• Centripetal Acceleration

  • Equivalent to: ( a_c = \frac{v^2}{R} )

• Free-Body Diagram (FBD):

  • Components: Tension (T), gravitational force (mg), necessary centripetal acceleration.

• Variables used in calculations will include mass (M), radius (R), velocity (v), and acceleration.

Highway Curves: Banked and Unbanked

Unbanked Curves

• Force Requirement: When a car goes around a curve, a net inward force must be present.

• Friction Role: If road is flat, friction is the force that provides this centripetal force.

If Friction is Insufficient

• Outcome: The car will attempt to move in a straight line rather than following the curve, which can be evidenced by skid marks.

Banked Curves

• Purpose of Banking: Helps to reduce skidding by allowing the centripetal force to be supplied by the horizontal component of the normal force.

• Ideal Speed on Banked Curve: Each banked curve has a specific speed at which friction is unnecessary for maintaining circular motion.

Example: Banking Angle Calculation

• Part (a): Find formula for the banking angle for no friction around a curve with radius r at speed v.

• Part (b): Calculate angle for a 50m radius curve at a speed of 50km/h.

  • Example results to include tangent relationships and accelerations.