Circular motion

Start with a free-body diagram containing the actual interactions. Then identify their components towards the centre and equate the radial resultant to mv²/r. For a satellite, gravity supplies it; for a car on a level bend, static friction can supply it. In a vertical circle, weight may assist or oppose the inward tension depending on position. If you draw both tension and a separate centripetal arrow as additional physical forces, you count the inward effect twice. Non-uniform circular motion can also have a tangential resultant that changes speed.

Compare velocity arrows at two nearby points on a circle. Their magnitudes are equal, but their difference points approximately inward; taking the limit of a short interval gives the radial acceleration. This acceleration is perpendicular to instantaneous velocity, so it changes direction without increasing kinetic energy. If the inward force disappears, the object initially moves along a tangent rather than radially outward. Other forces may then curve that path, as gravity does for a released object near Earth. Constant speed is therefore not the same condition as constant velocity.

A string can pull but cannot push. At a speed below the limiting value, maintaining the circular path would require a negative tension, so the taut-string model fails. A rigid rod can exert a force in either direction, so the same minimum-speed condition does not automatically apply. Likewise, a vehicle’s contact with the inside or outside of a track changes which normal-force direction is possible. Always draw the forces at the actual position. The speed at the bottom needed to reach this top condition also depends on the energy transferred during the ascent.

Choose inward as positive separately at each point. At the bottom, Tbottom − mg = mv²/r; at the top, Ttop + mg = mv²/r. In a freely moving system the speeds at the two positions are usually not equal, because gravitational potential energy changes. Then substitute the appropriate local speed into each equation rather than using the equal-speed difference blindly. This is a good example of a sign convention that changes direction around the path while the actual weight arrow remains downward everywhere.

One revolution is a complete turn of 2π radians, while a radian is defined by an arc length equal to the radius. The formulas a = rω² and v = rω assume radian measure because that definition links arc length directly to angle. Using the numerical rpm value as ω can produce an enormous squared error in force. Frequency in hertz and angular frequency are related but are not numerically interchangeable. Keep units beside each conversion, and check whether a quoted diameter needs halving before it is used as r.

The two expressions a = v²/r and a = rω² are consistent because v = rω. They answer different controlled comparisons. A car taking a wider bend at the same road speed needs less lateral force. Points farther from the centre of a rigid rotating disc share angular speed, but have greater tangential speed and acceleration. In an experiment, changing radius may unintentionally change rotation rate or tension, so identify the measured and controlled variables before describing a proportional relationship. A formula without its fixed quantities is not a complete prediction.