Circular Motion Lecture Notes

Circular Motion

Introduction to Circular Motion

• Circular motion is a two-dimensional motion in a plane with a fixed radius.
• Examples include a ball on a roulette wheel, a satellite in orbit, planets around a star, a merry-go-round, and a ball on the end of a string.

Uniform Circular Motion

• Uniform circular motion is the motion of an object traveling at a constant (uniform) speed on a circular path.
• Time Period (T): The time required for the object to travel once around the circle.

Speed of Uniform Circular Motion

• The speed v of an object in uniform circular motion is given by: v = \frac{2\pi r}{T} where:
  • r is the radius of the circular path.
  • T is the time period.
  • 2\pi r is the circumference of the circle, representing the total distance covered in one revolution.

Example Question 1

• A car wheel has a radius of 0.29 m and rotates at 830 revolutions per minute (rpm).
  • a) Find the time period T of the wheel.
  • b) Determine the speed (in m/s) at which the outer edge of the wheel is moving.

Velocity in Uniform Circular Motion

• Question: Is the velocity constant in uniform circular motion?
• Answer: No.
• Explanation: Although the speed is constant, the direction is always changing. Since velocity is a vector quantity (having both magnitude and direction), a change in direction means the velocity is changing.

Centripetal Acceleration

• An object in circular motion is always changing direction, causing the velocity to change, which implies there is acceleration.
• If the object is speeding up or slowing down, there is a parallel component of the acceleration (a_\parallel) parallel to the velocity.
• In uniform circular motion, the speed is constant, so a_\parallel = 0.
• However, the direction of motion is continuously changing. The perpendicular component (a\perp) of the acceleration (perpendicular to the velocity) is not zero and is constant. This is the centripetal acceleration (ac), which always points towards the center of the circle.
• The magnitude of the centripetal acceleration is given by: a_c = \frac{v^2}{r} where:
  • v is the speed of the object.
  • r is the radius of the circular path.
• If v is constant:
  • As r increases, a_c decreases.
• If T is constant (which seems to be a typo in the original notes and should likely refer to v constant):
  • As r increases, ac increases (This statement seems incorrect based on the formula ac = \frac{v^2}{r}, assuming v is constant).

QuickCheck Questions on Centripetal Acceleration

• Question 2: A car travels around a curve at a steady 45 mph. Is the car accelerating?
  • Answer: Yes, because the direction is changing, hence the velocity is changing and acceleration is non-zero.
• Question 3: A car travels around a curve at a steady 45 mph. Which vector shows the direction of the car’s acceleration?
  • The acceleration vector should point towards the center of the curve.

Example: Bobsled Track

• Question 4 & 5: The bobsled track at the 1994 Olympics in Lillehammer, Norway, had turns with radii of 33 m and 24 m. Find the centripetal acceleration at each turn for a speed of 34 m/s.
  • What happens to the centripetal acceleration as the circle grows larger?
  • Centripetal acceleration for the smaller circle is greater.
  • Calculations:
    • For r = 33 m: a_c = \frac{(34 m/s)^2}{33 m} \approx 35.03 m/s^2
    • For r = 24 m: a_c = \frac{(34 m/s)^2}{24 m} \approx 48.17 m/s^2

Question 6: Centripetal Acceleration Comparison

• Consider a man standing at the pole and a man standing at the equator. Who has a larger centripetal acceleration?
  • Answer: Man at the equator. They have different radii of rotation due to Earth's shape.

Centripetal Force

• For the same speed, the tighter the circle/curve, the greater the centripetal acceleration.
• The centripetal force is the net force required to keep an object of mass m, moving at a speed v, on a circular path of radius r. Its magnitude is:
  Fc = mac = m\frac{v^2}{r}
• The centripetal force always points toward the center of the circle (same as the centripetal acceleration) and continually changes direction as the object moves.
• The phrase “centripetal force” does not denote a new and separate force created by nature, but rather labels the net force pointing toward the center of the circular path.
• This net force is the vector sum of all the force components that point along the radial direction (e.g., tension, friction, normal force, or gravitational force).

Questions on Centripetal Force

• Questions 7 & 8 & 9 & 10 & 11: A man holds a rope causing a model airplane (with mass m) to fly at a constant speed along a circular path parallel to the ground.
  • If the airplane were more massive, the force required to keep it moving at the same speed and radius would increase, according to F_c = m\frac{v^2}{r}.
  • If the airplane were to move faster, the force required to keep it moving with the same radius would increase, because F_c is proportional to v^2.
  • The centripetal force is caused by the tension in the rope.
  • The tension in the rope with length r and speed v is given by T = m\frac{v^2}{r}.
  • Comparing tensions for speeds v1 and v2 where v1 > v2, the tension is higher on the rope if the plane moves with velocity v_1, since T is proportional to v^2.