Vertical Circular Motion

What is Vertical Circular Motion?

Movement of a body in a circular path in the vertical plane.

Why is Vertical Circular Motion typically non-uniform? (Reason for Non-Uniform Motion)

Height changes -> GPE changes -> KE changes -> speed changes.

When can speed remain constant in Vertical Circular Motion?

If work is done by an external force on the system.

What forces are considered in Vertical Circular Motion?

Centripetal force (Fc = mac) and conservation of energy.

What is the equation for Tension at the TOP of a vertical circular path (speed constant)?

T_top + mg = mv²/r.

What is the equation for Tension at the BOTTOM of a vertical circular path (speed constant)?

T_bottom - mg = mv²/r.

What is the relationship between tension at the top and bottom of vertical circular motion?

Tension is smallest at the top and largest at the bottom.

What happens to GPE, KE, and speed during vertical circular motion (assuming constant speed)?

GPE changes continuously. KE and speed are kept constant by positive work done by external force moving up, and negative work done moving down.

What is the energy conservation equation for the whole system (Vertical Circular Motion with external work)?

Initial GPE + work done by external force = final GPE.

What is Non-uniform Vertical Circular Motion? (Speed not Constant)

Motion where speed changes continuously due to interchanges between KE and GPE (e.g., roller coaster without engine).

What are the forces acting on a carriage in non-uniform vertical circular motion?

Normal contact force (N) by track and its weight (mg).

What is the condition for a carriage to remain in contact with the track at the TOP of a loop?

Ntop ≥ 0, which means mvtop² / r ≥ mg, so v_top ≥ √rg.

What is the Minimum Speed at the TOP to remain in contact?

v_top,min = √rg.

What is the Minimum Speed at the BOTTOM to just make it over the top of the loop?

vbottom,min = √5gr (assuming negligible air resistance/friction, using conservation of energy: ½mvbottom,min² = ½mv_top,min² + mg(2r)).

What happens if the carriage speed is larger than √5gr at the bottom?

It will remain well in contact throughout the loop.

What are the components of the resultant force in non-uniform vertical circular motion?

A centripetal (radial) component (responsible for circular motion) and a tangential component (causes speed to vary).