Applications of circular motion
Section: Motion in a Circle | Syllabus: Cambridge A Level Physics (9702)
Banked Turns
On high-speed tracks (velodromes or race circuits), bends are "banked" at an angle θ. This reduces reliance on friction.
- The normal force N provides a horizontal component toward the centre: N sin θ.
- The vertical component supports the weight: N cos θ = mg.
Condition for zero friction:
tan θ = (v^2/rg) Diagram: Banked Track ForcesShow a car on a slope of angle θ. Draw the Normal force N at an angle to the vertical. Resolve N into N cos θ (vertical) and N sin θ (horizontal/centripetal).
Vertical Circular Motion
Unlike horizontal motion, gravity acts along the radius at the top and bottom, changing the net force.
Force Analysis (Uniform Speed)- At the Top (Point 1): Both T and W point down. F_c = T + mg.
- At the Bottom (Point 3): T points up, W points down. F_c = T - mg T = F_c + mg.
Conclusion: The string (or seat) must provide the maximum force at the bottom. This is where a string is most likely to break and where passengers feel "heaviest".
Minimum Speed at the Top
To stay in a circle (e.g., water in a bucket), the weight mg must be less than or equal to the required centripetal force.
v_min = √(rg)The Conical Pendulum
A mass whirled in a horizontal circle by a string at an angle θ to the vertical.
- T sin θ = F_c (Horizontal component provides rotation).
- T cos θ = mg (Vertical component supports weight).
Humps and Bridges
When a car drives over a curved bridge of radius r, the weight acts toward the centre, and the normal force N acts outward.
F_c = mg - N N = mg - (mv^2/r)If the car goes too fast (v = √(rg)), the normal force N becomes zero, and the car leaves the road.
Work Done in Circular MotionThe work done by the centripetal force is always zero because the force is always perpendicular to the direction of motion (W = Fd cos 90^). Therefore, the kinetic energy (and speed) of the object remains constant.
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