Circular motion in practice: horizontal and vertical circles
Cars on bends, a conical pendulum, and a ball on a string in a vertical circle: the forces at the top and the bottom.
A roller-coaster at the bottom of a dip pushes you hard into the seat. At the top of a hill you feel light. Same speed, same rider: what changes?
A 0.20 kg puck moves at 3.0 m s⁻¹ on a frictionless table, held by a string of length 0.60 m. The tension is…
A car goes round a flat bend at a speed that is doubled. The friction needed to keep it on the bend becomes…
A 0.50 kg ball is whirled in a vertical circle of radius 1.0 m. At the bottom its speed is 6.0 m s⁻¹. The tension at the bottom is (g = 9.81 m s⁻²)…
The slowest speed at the top of a vertical circle of radius 0.90 m, with a string that must stay taut, is about…
A car goes over the top of a hill of radius 40 m at 15 m s⁻¹. The normal force on a 60 kg passenger is (g = 9.81 m s⁻²)…
A ball on a string moves in a vertical circle. Explain, using forces, why the string is most likely to break at the bottom of the circle.
The centripetal force is…
Static friction has a maximum of…
Key points
- • Horizontal circle: the resultant force is horizontal, towards the centre: T, friction or a component supplies mv²/r.
- • Flat bend: v_max = √(μ_s g r). Conical pendulum: tan θ = v² / (rg).
- • Vertical circle, bottom: T = m(g + v²/r). Top: T = m(v²/r − g). Taut string at the top needs v ≥ √(gr).
- • Non-uniform motion: the resultant has a tangential part (speed) and a radial part (direction).
Formulas
- T − mg = mv²/r (bottom)
- T + mg = mv²/r (top)
Key terms
- Apparent weight :
- the normal force you feel; greater than mg at the bottom of a dip, less at the top of a hill
Common errors
- • Adding the weight at the bottom instead of subtracting it
- • Using the centripetal equation at points other than top and bottom
- • Treating banked-road problems quantitatively (not required)