Circular Motion

A road of radius 20 m is banked at an angle whose tangent is 0.75. The coefficient of friction between the tyres and the road is 0.5. Find the maximum and minimum speeds at which a vehicle can safely negotiate the curve. (g = 10 m/s²)

A 1 kg bob moves in a vertical circle of radius 2 m on a string, with speed 10 m/s at the lowest point. Find the tension in the string at the instant the string is exactly horizontal. (g = 10 m/s²)

A conical pendulum has a string of length 2 m making an angle of 37° with the vertical. Find the speed of the bob and the time period of revolution. (g = 10 m/s², sin37° ≈ 0.6, cos37° ≈ 0.8, tan37° ≈ 0.75)

Two particles move on the same circle of radius 2 m, one at speed 4 m/s clockwise and the other at speed 6 m/s anticlockwise. Find the rate at which the angle between them (measured the short way) is closing.

A ball on a string of length 1.6 m is given a speed of 8 m/s at the lowest point. Determine whether the ball completes the full circle, oscillates, or leaves the circular path partway. (g = 10 m/s²)

A vertical circular loop is set up inside a trolley accelerating horizontally at a = g (numerically equal to g). Find the effective gravity magnitude and its direction relative to the true vertical, as felt inside the trolley.