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. The diagram shows the velocity vector tangent to the circle and the centripetal acceleration pointing inward.

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 $F_c \propto m$).

Examples in context

Example 1. Bathurst 1000 cornering at Forrest's Elbow. A V8 Supercar of mass $m = 1410 \text{ kg}$ rounds the tight right-hander at $v = 22 \text{ m/s}$ on a radius $r = 38 \text{ m}$. The required centripetal force is $F_c = m v^2 / r = 1410 \times 22^2 / 38 = 17{,}960 \text{ N}$, supplied by static friction between the slicks and the bitumen. Comparing to the car's weight $m g = 1410 \times 9.8 = 13{,}820 \text{ N}$ gives a required coefficient $\mu \geq F_c / (m g) = 1.30$, achievable only on a hot dry track with race slicks. In the wet, $\mu \approx 0.6$ caps cornering speed at $v = \sqrt{\mu g r} = \sqrt{0.6 \times 9.8 \times 38} = 15.0 \text{ m/s}$.

Example 2. Conical pendulum demo at Sydney Observatory. A $0.50 \text{ kg}$ pendulum bob on a $1.20 \text{ m}$ string swings in a horizontal circle with the string at $25^{\circ}$ to vertical. The radius of the circle is $r = L \sin 25^{\circ} = 0.507 \text{ m}$. Tension supplies both the vertical balance ($T \cos 25^{\circ} = m g$) and the centripetal force ($T \sin 25^{\circ} = m v^2 / r$). Dividing gives $v = \sqrt{g r \tan 25^{\circ}} = \sqrt{9.8 \times 0.507 \times 0.466} = 1.52 \text{ m/s}$. Period is $T_p = 2 \pi r / v = 2.10 \text{ s}$. Heritage demos at the Observatory let visitors check the prediction against a stopwatch.

Try this

Q1. State the direction of the centripetal acceleration of an object in uniform circular motion and write the equation linking $a_c$, $v$ and $r$. [2 marks]

• Cue. Toward the centre of the circle; $a_c = v^2/r$. One mark each.

Q2. A $750 \text{ kg}$ car rounds an unbanked roundabout of radius $25 \text{ m}$ at $14 \text{ m/s}$. Calculate the minimum coefficient of static friction needed and comment on whether typical dry bitumen ($\mu_s \approx 0.85$) suffices. [4 marks]

• Cue. $\mu \geq v^2 / (g r) = 14^2 / (9.8 \times 25) = 0.80$. Just below 0.85, so dry bitumen is safe but wet conditions are not.

Q3. A satellite orbits Earth in a circle. (a) Identify the real force providing the centripetal force. (b) Derive the orbital speed $v$ in terms of $G$, $M_E$ and $r$. (c) Show that doubling $r$ reduces $v$ by a factor $\sqrt{2}$. [1+3+2 marks]

• Cue. (a) Gravitational attraction. (b) Equate $G M m / r^2 = m v^2 / r$, solve for $v$. (c) $v \propto 1/\sqrt{r}$, so $r \to 2r$ gives $v \to v/\sqrt{2}$.