Conduct investigations to explain and evaluate, for objects executing uniform circular motion, the relationships that exist between centripetal force, mass, speed and radius, and solve problems using the relationships a_c = v^2 / r, v = 2 pi r / T, F_c = m v^2 / r and omega = delta theta / delta t

What this dot point is asking

NESA wants you to model an object moving in a circle at constant speed, derive the relationships between centripetal acceleration, force, mass, speed and radius, and apply them in calculations. You also need to identify what physical force provides the centripetal force in a given situation (gravity, friction, tension, normal force, or a combination).

The answer

An object in uniform circular motion travels in a circle of radius $r$ at constant speed $v$. Even though speed is constant, the velocity vector is constantly changing direction, so the object is accelerating.

Centripetal acceleration

The acceleration points toward the centre of the circle:

Centripetal force

By Newton's second law, this acceleration requires a net inward force:

Centripetal force is not a new kind of force. It is whatever real force happens to be acting toward the centre of the circle: gravity for a satellite, friction for a car on a flat bend, tension for a ball on a string, the normal force component for a car on a banked road.

Period, speed and angular velocity

The period $T$ is the time for one full revolution. In one period the object travels the circumference $2\pi r$:

The angular velocity $\omega$ is the rate at which the angle swept changes:

So $v = \omega r$ and $a_c = \omega^2 r$.

Relationships between variables

For a given object on a circular path:

• Doubling the speed quadruples the centripetal force (because $F_c \propto v^2$).
• Doubling the radius halves the centripetal force at the same speed (because $F_c \propto 1/r$).
• Doubling the mass doubles the centripetal force (because $F_c \propto m$).

Worked example with numbers

A $0.50$ kg ball is whirled in a horizontal circle on the end of a $1.2$ m string at $3.0$ revolutions per second. Find the speed, the centripetal acceleration, and the tension in the string.

Period: $T = \frac{1}{3.0} = 0.333$ s.

Speed: $v = \frac{2 \pi r}{T} = \frac{2 \pi \times 1.2}{0.333} = 22.6$ m/s.

Centripetal acceleration: $a_c = \frac{v^2}{r} = \frac{22.6^2}{1.2} = 426 \text{ m/s}^2$.

Tension (the source of the centripetal force, assuming a horizontal circle): $T_{\text{string}} = \frac{m v^2}{r} = 0.50 \times 426 = 213$ N.

Try it: Centripetal force calculator - enter mass, speed and radius and get F, a, T and ω.

Common traps

Calling centripetal force a separate force. It is not. It is the net force directed toward the centre, supplied by friction, gravity, tension, or normal force. In a free-body diagram, you draw the real forces (gravity, normal, tension) and show that their resultant points to the centre.

Confusing centripetal and centrifugal. Centrifugal force is a fictitious force that appears only in a rotating reference frame. In HSC, work in the ground frame and use centripetal force only.

Forgetting the direction. Velocity is tangential to the circle. Acceleration and net force are radial, pointing toward the centre.

Mixing units of angular velocity. Use radians per second for $\omega$, not revolutions per second or degrees per second, unless you explicitly convert.

Using diameter instead of radius. $a_c = v^2 / r$, not $v^2 / d$.

In one sentence

Uniform circular motion is constant-speed motion along a circular path, in which a centripetal acceleration $a_c = v^2/r$ directed toward the centre is produced by a net inward force $F_c = m v^2 / r$ supplied by whatever real force (gravity, friction, tension, normal) acts radially.