Investigate centripetal force, acceleration, radius, period, and speed

Ontario Grade 12 Physics

Dynamics

Centripetal force, acceleration, radius, period, and speed

An object moving in a circle can have a steady speed and still be accelerating. Its direction changes continuously, so its velocity changes. In this lesson, you will connect that inward acceleration to the net force on the object and use period, radius, and speed to describe the motion. You will also consider how an investigation could test these relationships without treating planned or simulated results as measured evidence.

What you will learn

Describe circular motion using a clear system, reference frame, and direction convention.
Relate centripetal acceleration, speed, radius, and period.
Explain how the net inward force produces centripetal acceleration.
Plan an investigation that tests relationships among the quantities in circular motion.

1. Start with the system and motion

A physical system is the object or group of objects being studied. For a simple example, take a small object moving in a horizontal circle. Use a reference frame fixed to the room, so the object's changing position is described from the room's point of view. At each instant, choose the direction toward the centre of the circle as positive inward. The inward direction changes as the object moves around the circle.

A scalar has size only. Speed, radius, and period are scalars. A vector has both size and direction. Velocity, acceleration, and force are vectors. In circular motion, the velocity points along the tangent to the path. The tangent is the direction the object is moving at that instant. The centripetal acceleration points toward the centre, at right angles to the instantaneous velocity.

Centripetal means centre-seeking. Centripetal force is not a new kind of force. It is the name for the net force directed toward the centre that keeps the object moving along a circular path. Depending on the situation, ordinary forces such as tension, friction, or gravity can contribute to this net inward force.

\vec{a}_c points toward the centre

Velocity is tangent to the circular path; centripetal acceleration points inward.
Centripetal force means the net inward force, not an extra force to add to a force diagram.

2. Connect radius, period, speed, and acceleration

The radius, r, is the distance from the centre of the circular path to the moving object. One complete trip around the circle covers a distance equal to its circumference, 2πr. The period, T, is the time for one complete trip. Period is measured in seconds. Speed is distance travelled per time and is measured in metres per second.

For steady circular motion, speed equals the circumference divided by the period. The inward acceleration depends on speed and radius: a faster object needs more inward acceleration, while a larger circle needs less acceleration at the same speed. Acceleration is measured in metres per second squared.

The relationship can also be written using period. If the radius stays fixed, a shorter period means a faster trip and a greater inward acceleration. These equations describe the size of the acceleration. Its direction is still toward the centre.

v=\frac{2\pi r}{T},\quad a_c=\frac{v^2}{r}=\frac{4\pi^2r}{T^2}

Use metres for radius, seconds for period, metres per second for speed, and metres per second squared for acceleration.
The period is the time for one revolution, not the time for a fraction of a revolution.

3. Relate acceleration to net inward force

Newton's second law links net force and acceleration. For circular motion, the net force toward the centre produces the centripetal acceleration. The object's mass is measured in kilograms, and force is measured in newtons. One newton is equivalent to one kilogram metre per second squared.

A free-body diagram should show the actual forces acting on the chosen object. Resolve those forces along the inward direction and find their net inward component. That net component is the centripetal force. Do not draw “centripetal force” as an additional force if the diagram already includes the forces causing the inward result.

A useful investigation varies one factor at a time while keeping the others as steady as practical. For example, a proposed investigation could keep the radius and moving object the same, vary the speed, and compare the calculated inward acceleration or required net force. A timer and a measured radius can provide evidence for period and radius; a speed can then be calculated from those measured quantities. Such a procedure is a plan, not a claim that measurements have already been collected. Record actual measurements separately from predictions or simulated values.

F_c=ma_c=\frac{mv^2}{r}

Centripetal force is the net inward force required for the circular motion.
Keep measured evidence distinct from calculated predictions and proposed procedures.

4. Use the relationships to interpret changes

The equations help predict how changing one quantity affects another, provided the remaining quantities named in the comparison are held constant. At fixed radius, doubling speed makes the centripetal acceleration four times as large because speed is squared. At fixed speed, doubling radius makes the acceleration half as large. At fixed radius, halving the period doubles the speed and makes the acceleration four times as large.

These comparisons are useful when planning an investigation and when checking whether a result is reasonable. If an object travels around the same circle in less time, its speed must increase. The required inward acceleration also increases. Always state what is being held constant; without that information, a comparison may not have a single answer.

a_c\propto v^2\text{ at fixed }r

At fixed radius, acceleration is proportional to speed squared.
At fixed speed, acceleration decreases when radius increases.

Circular-motion quantities

Quantity	Meaning	SI unit
Radius, r	Distance from centre to object	m
Period, T	Time for one complete revolution	s
Speed, v	Distance travelled per time	m/s
Centripetal acceleration, a_c	Acceleration directed toward the centre	m/s²
Centripetal force, F_c	Net force directed toward the centre	N

Worked example

Find speed and acceleration from a period

A marker moves steadily around a circle of radius 0.80 m. One revolution takes 2.0 s. Find its speed and centripetal acceleration.

Define the motion
The system is the marker, viewed from a room-fixed frame. Choose inward as positive at the marker's current position. The unknowns are speed and the magnitude of centripetal acceleration.
Find speed
One revolution covers the circumference. Divide that distance by the period, keeping metres and seconds in the substitution.
$v=\frac{2\pi r}{T}=\frac{2\pi(0.80\,\mathrm{m})}{2.0\,\mathrm{s}}=2.5\,\mathrm{m/s}$
Find inward acceleration
Use the speed and radius. The acceleration points toward the centre, so its direction is inward.
$a_c=\frac{v^2}{r}=\frac{(2.5\,\mathrm{m/s})^2}{0.80\,\mathrm{m}}=7.8\,\mathrm{m/s^2}$

Answer: The marker's speed is 2.5 m/s, and its centripetal acceleration is 7.8 m/s² inward.

Check: The acceleration unit reduces to m/s². A shorter period or smaller radius at this speed would require greater inward acceleration, so the result is consistent with the relationships.

Worked example

Find the required net inward force

A 0.50 kg object moves at 3.0 m/s in a circle of radius 1.5 m. Find the net centripetal force.

Define the system
The system is the moving object in a room-fixed frame. Inward is positive. The known values are mass, speed, and radius; the unknown is the net inward force.
Calculate acceleration
First determine the inward acceleration from speed and radius. Its direction is toward the circle's centre.
$a_c=\frac{v^2}{r}=\frac{(3.0\,\mathrm{m/s})^2}{1.5\,\mathrm{m}}=6.0\,\mathrm{m/s^2}$
Apply Newton's second law
The net inward force equals mass times inward acceleration. Keep the direction with the result.
$F_c=ma_c=(0.50\,\mathrm{kg})(6.0\,\mathrm{m/s^2})=3.0\,\mathrm{N}$

Answer: The required net centripetal force is 3.0 N inward.

Check: Kilograms multiplied by metres per second squared give newtons. The answer is a net force toward the centre, not necessarily the size of any one force acting on the object.

Worked example

Compare two periods at the same radius

An object moves in a circle of radius 0.60 m. Its period changes from 1.5 s to 1.0 s. Find its speed and acceleration before and after the change.

Set the comparison
The system is the same object in the same room-fixed frame. Inward is positive at each position. Radius is held constant, and the two periods are given.
Calculate the initial values
Use the circumference divided by period for speed, then use speed squared divided by radius for acceleration.
$v_1=\frac{2\pi(0.60\,\mathrm{m})}{1.5\,\mathrm{s}}=2.5\,\mathrm{m/s},\quad a_1=\frac{(2.5\,\mathrm{m/s})^2}{0.60\,\mathrm{m}}=10\,\mathrm{m/s^2}$
Calculate the later values
Repeat with the shorter period. The shorter time for the same circumference should produce a larger speed and acceleration.
$v_2=\frac{2\pi(0.60\,\mathrm{m})}{1.0\,\mathrm{s}}=3.8\,\mathrm{m/s},\quad a_2=\frac{(3.8\,\mathrm{m/s})^2}{0.60\,\mathrm{m}}=24\,\mathrm{m/s^2}$

Answer: Initially, speed is 2.5 m/s and acceleration is 10 m/s² inward. Later, speed is 3.8 m/s and acceleration is 24 m/s² inward.

Check: The period falls to two-thirds of its initial value, so speed rises by a factor of 1.5 and acceleration rises by a factor of about 2.25. The calculated values show these expected changes within rounding.

Common mistakes and how to avoid them

Treating centripetal force as an extra force in addition to tension, friction, or other forces.

Correction: Draw the actual forces. Their net inward component is the centripetal force.

Drawing velocity toward the centre.

Correction: Velocity is tangent to the path. Centripetal acceleration and net centripetal force point inward.

Using the diameter as the radius.

Correction: Radius is the centre-to-path distance. If given the diameter, divide it by two.

Using the period as though it were speed.

Correction: Period is time per revolution. Find speed by dividing the circumference by the period.

Lesson summary

Circular motion involves changing velocity direction, even when speed is steady.
Centripetal acceleration and net centripetal force point toward the circle's centre.
Speed is circumference divided by period; acceleration depends on speed squared and radius.
Use stated conditions, SI units, and measured evidence carefully when investigating relationships.

Check your understanding

Question 1

At the same speed, what happens to centripetal acceleration if the radius is doubled?

It doubles.
It is halved.
It becomes four times as large.
It does not change.

Show answer and explanation

It is halved.

At fixed speed, centripetal acceleration is speed squared divided by radius. Doubling the radius halves the acceleration.

Question 2

Which way does centripetal acceleration point at an instant during circular motion?

Along the direction of motion
Away from the centre
Toward the centre
Opposite the velocity and away from the centre

Show answer and explanation

Toward the centre

Centripetal acceleration points inward, toward the centre. The velocity is tangent to the path.

Question 3

A circle has radius 0.50 m and period 2.0 s. What is the approximate speed?

0.79 m/s
1.6 m/s
3.1 m/s
6.3 m/s

Show answer and explanation

0.79 m/s

The circumference is about 3.14 m. Dividing by 2.0 s gives about 1.6 m/s—wait: the correct calculation is 2π(0.50 m)/2.0 s = 1.57 m/s. Therefore the correct option is 1.6 m/s.

Key terms

Centripetal: Directed toward the centre of a circular path.
Period: The time for one complete revolution.
Radius: The distance from the centre of a circle to its path.
Scalar: A quantity with size but no direction.
Vector: A quantity with both size and direction.

Continue through SPH4U

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Physics (SPH4U), expectation B2.7. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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