Key idea: For a given radius, the centripetal force required to keep an object moving in a circle is \$F = \dfrac{mv^2}{r}\$.
If the mass \$m\$ and radius \$r\$ stay the same, increasing the force \$F\$ must be balanced by a larger speed \$v\$.
Analogy: Imagine a marble on a flat track that you push harder. The harder you push, the faster the marble rolls around the loop.
Quick formula: \$v = \sqrt{\dfrac{Fr}{m}}\$ – more \$F\$ → larger \$v\$.
Key idea: With fixed mass \$m\$ and speed \$v\$, the required centripetal force is \$F = \dfrac{mv^2}{r}\$.
If \$F\$ grows while \$m\$ and \$v\$ stay the same, the only way to satisfy the equation is to reduce \$r\$.
Analogy: Picture a car on a circular track. If you increase the steering force (turning harder), the car takes a tighter turn – the radius of the circle shrinks.
Quick formula: \$r = \dfrac{mv^2}{F}\$ – more \$F\$ → smaller \$r\$.
Key idea: If you want the same speed \$v\$ and radius \$r\$ but the mass \$m\$ becomes larger, the centripetal force must increase proportionally: \$F = \dfrac{mv^2}{r}\$.
Analogy: Think of a child on a swing. If the child gets heavier, you need to pull harder (apply more force) to keep the swing moving at the same speed and in the same circular path.
Quick formula: \$F \propto m\$ – double the mass → double the force (for constant \$v\$ and \$r\$).
| Scenario | Variables Held Constant | Effect of Increasing Force | Key Formula |
|---|---|---|---|
| Speed ↑ | \$m\$, \$r\$ constant | \$v\$ increases | \$v = \sqrt{\dfrac{Fr}{m}}\$ |
| Radius ↓ | \$m\$, \$v\$ constant | \$r\$ decreases | \$r = \dfrac{mv^2}{F}\$ |
| Force ↑ (to keep \$v\$, \$r\$) | \$v\$, \$r\$ constant, \$m\$ ↑ | \$F\$ increases proportionally to \$m\$ | \$F = \dfrac{mv^2}{r}\$ |