Think of the roller coasters you enjoy at the amusement park. They are fast, thrilling, and move in loops - vertical circles. We are going to dive deep into the physics of motion in a vertical circle and understand the forces acting on objects.

Moving in a Vertical Circle

When a mass is moving in a vertical circle, the forces acting on it change depending on its position. Picture a yo-yo being swung in a circle - that's our scenario here.

• At Point A (when the string is horizontal): The tension in the string is the horizontal centripetal force towards the centre of the circle. The weight of the object acts downwards due to gravity.
• At Point B (top of the circle): The tension in the string and the weight both act downwards. Therefore, the tension required is less than when the string is horizontal. The equation for the tension at the top is:
• T_down = m* \(\frac{v^2}{r}\) - mg
• At Point C (bottom of the circle): The tension and the weight act vertically but in opposite directions. At the bottom, the tension in the string needs to overcome the weight of the object and provide the required centripetal force. The equation for the tension at the bottom is

T^up = m*\(\frac{v^2}{r}\) + mg

At the top of the circle, the tension is at its minimum, while at the bottom, it is at its maximum. So if you've ever wondered why a yo-yo string is more likely to snap when the yo-yo is at the bottom of its circle, now you know!

Practical Applications

To bring this theory to life, imagine a car moving over a hump-shaped bridge. If the bridge's shape is part of a circle, the car will lose contact with the bridge at the top if its speed equals √gr (g is gravity, r is radius of curvature). This speed is the maximum safe speed for the car. Any faster, and the car becomes airborne – fun in a movie, but dangerous in real life.

• Hammer Thrower: A hammer thrower swings the hammer (mass = 4.0kg, radius = 2.1m) 7.5 times in 5.2 seconds before releasing it.
•  Average angular speed of the hammer = 15π rad  5.2 = 9.1 rads^-1
• The average tension in the chain = centripetal force required for rotation = mrω² = 4.0 * 2.1 * 9.1² = 690N
• Stone and String: A stone (mass = 0.25kg) is attached to a string and moves in a vertical circle (radius = 0.80m) at a constant speed. The string will break if the tension exceeds 10N.
• The string is more likely to break when the stone passes the lowest point (Point C) as this is where the tension is at its maximum.
• Car over a Bridge: A car (mass = m) moves at speed v over a bridge whose central part can be modelled as a section of a circle of radius r.

Derive an expression, in terms of m, v and r, for the magnitude of the normal reaction force between the car and the bridge as the car passes the top. It's given that r = 60m and m =1400kg.

Remember, as the car moves at a speed that allows it to just lose contact with the bridge, the normal force becomes zero.