Motion in a Circle
Newton’s Laws and Circular Motion
Objectives:
- Establish expressions for angular velocity, angular acceleration, and centripetal force.
- Solve numerical problems involving motion in a circle.
- Interpret the area under a force–time graph in terms of impulse.
- Deduce the relationship between linear speed and angular speed.
- Apply concepts of circular motion in collisions (elastic and inelastic) where applicable.
Key Formulas and Concepts
Angular Displacement: θ = S / R
(θ in radians, S = arc length, R = radius)
Angular Velocity: ω = θ / t
(ω in radians per second)
Relationship between Linear Speed and Angular Speed:
S = R * θ so that v = S / t = R * (θ / t) = ω * R
v = ω * R
Angular Acceleration: α = Δω / t
(α in radians per second squared)
Centripetal Force: Fc = m * v² / R = m * ω² * R
(Force required to keep an object moving in a circle)
Centrifugal Force: A perceived force acting outward on a body moving in a circle; not a real force but an effect of inertia in a rotating reference frame.
Worked Examples
The following 6 worked examples (each in a green box) illustrate the application of the above formulas and concepts.
Example 1: A stone tied to a string moves in a circle of radius 2 m. If its angular velocity is 4 rad/s, find its linear speed.
Solution: v = ω * R = 4 * 2 = 8 m/s.
Example 2: A wheel rotates with an angular displacement of 6 radians in 2 s. Calculate its angular velocity.
Solution: ω = θ / t = 6/2 = 3 rad/s.
Example 3: A car of mass 1000 kg is moving in a circular track of radius 50 m at 20 m/s. Determine the centripetal force acting on the car.
Solution: Fc = m * v² / R = 1000 * (20²)/50 = 1000 * 400/50 = 8000 N.
Example 4: A merry-go-round accelerates from rest to an angular velocity of 2 rad/s in 5 s. Find its angular acceleration.
Solution: α = Δω / t = 2/5 = 0.4 rad/s².
Example 5: A cyclist travels around a circular track of radius 30 m at a constant speed of 12 m/s. Calculate the angular velocity of the cyclist.
Solution: ω = v / R = 12/30 = 0.4 rad/s.
Example 6: In a collision problem, a 2 kg ball moving at 3 m/s strikes a stationary 3 kg ball elastically. Using conservation of momentum and kinetic energy, the final velocities can be computed (detailed steps omitted here for brevity).
Note: For elastic collisions, use:
v₁' = [(m₁ - m₂)/(m₁ + m₂)]v₁ + [2m₂/(m₁ + m₂)]v₂ and v₂' = [2m₁/(m₁ + m₂)]v₁ + [(m₂ - m₁)/(m₁ + m₂)]v₂.
Comments
Post a Comment
We’d love to hear from you! Share your thoughts or questions below. Please keep comments positive and meaningful, Comments are welcome — we moderate for spam and civility; please be respectful.