Circular Motion: quick revision, 19 bites

The Core Insight: Acceleration Points Inward

Velocity changes direction, not speed.

In uniform circular motion, speed is constant but **velocity** is not, because direction changes continuously. This change in velocity over time is an **acceleration**. The direction of this acceleration is always toward the center of the circle, hence the name **centripetal** (center-seeking) acceleration.

Takeaway. Centripetal acceleration is the rate of change of velocity direction, always pointing toward the circle's center.

Defining Centripetal Acceleration

A precise definition for a precise concept.

**Centripetal acceleration** ($a_c$) is the acceleration experienced by an object moving in a circular path at constant speed. It is always directed radially inward, toward the center of the circle, and is responsible for changing the direction of the velocity vector.

Takeaway. Centripetal acceleration is the inward acceleration that keeps an object on a circular path.

Geometric Derivation: Step 1

Start with two velocity vectors.

Consider an object moving with constant speed $v$ on a circle of radius $r$. At a time $t_1$, its velocity is $\vec{v}_1$, and after a small time interval $\Delta t$, at time $t_2$, its velocity is $\vec{v}_2$. Both vectors have the same magnitude $v$, but different directions.

Takeaway. The change in velocity is found by vector subtraction of the two velocity vectors.

Geometric Derivation: Step 2

The triangle of velocities is similar.

The angle between $\vec{v}_1$ and $\vec{v}_2$ is the same as the angle $\Delta \theta$ swept out by the object at the center of the circle, because the velocity is always tangent to the circle.

Takeaway. Similar triangles relate the velocity change to the geometry of the circle.

Geometric Derivation: Step 3

Use similarity to find Δv.

From the similarity of the triangles, the ratio of corresponding sides is equal. The side of the velocity triangle corresponding to the chord length $\Delta s$ is $\Delta v$. The side corresponding to the radius $r$ is the speed $v$.

Takeaway. The ratio of velocity change to speed equals the ratio of arc length to radius.

Geometric Derivation: Step 4

Divide by time to get acceleration.

Rearrange the equation to solve for $\Delta v$: $\Delta v = \frac{v \cdot \Delta s}{r}$. Now, divide both sides by the time interval $\Delta t$ to get the acceleration.

Takeaway. Dividing by the time interval transforms the velocity change into an acceleration.

The Final Formula

The result of the derivation.

As $\Delta t$ approaches zero, the average acceleration becomes the instantaneous acceleration, and $\frac{\Delta s}{\Delta t}$ becomes the instantaneous speed $v$. This yields the final formula for **centripetal acceleration**.

Takeaway. The magnitude of centripetal acceleration is $v^2/r$.

Direction: Always Inward

The key to the direction.

In the limit as $\Delta t \to 0$, the angle $\Delta \theta$ also approaches zero. The vector $\Delta \vec{v}$ becomes perpendicular to the velocity vector $\vec{v}$. Since the velocity is tangent to the circle, the perpendicular direction points **radially inward** toward the center.

Takeaway. The direction of centripetal acceleration is always toward the center of the circle.

Alternative Form: Angular Velocity

Express in terms of ω.

Since the speed $v$ is related to the angular velocity $\omega$ by $v = \omega r$, the centripetal acceleration can also be written in terms of $\omega$.

Takeaway. Centripetal acceleration can be expressed as $\omega^2 r$.

Real-World Application: Car Turning

Why you feel pushed sideways.

When a car turns a corner at constant speed, it experiences **centripetal acceleration** directed toward the center of the turn. The friction between the tires and the road provides the necessary **centripetal force** to produce this acceleration.

Takeaway. Centripetal acceleration explains why cars need friction to turn corners safely.

Real-World Application: Satellites in Orbit

Gravity provides the inward pull.

A satellite orbiting Earth moves in a circular path (approximately). Its **centripetal acceleration** is provided by the gravitational force between the satellite and Earth. This force continuously pulls the satellite toward Earth's center, changing its direction without changing its speed.

Takeaway. Gravitational force provides the centripetal acceleration for orbiting satellites.

Common Mistake: Confusing Direction

Outward is not correct.

A frequent error is thinking that centripetal acceleration points **outward** from the center. This confusion arises from the feeling of being pushed outward when turning in a car. However, this feeling is due to **inertia** — your body wants to continue in a straight line.

Takeaway. Centripetal acceleration points inward, not outward.

Exam Tip: Know Both Formulas

Be ready for either form.

Takeaway. Choose the formula based on whether you have linear speed or angular velocity.

Memory Trick: 'v-squared over r'

A simple phrase to remember.

To remember the formula, think: **'v-squared over r'** — the speed squared divided by the radius. The units work out to m/s², which is acceleration.

Takeaway. Say 'v-squared over r' to recall the centripetal acceleration formula.

Did You Know? Centripetal vs. Centrifugal

Two forces, one real.

**Centripetal force** is a real force that acts toward the center of a circle, causing centripetal acceleration. **Centrifugal force** is a fictitious force that appears to push objects outward in a rotating reference frame.

Takeaway. Centripetal force is real; centrifugal force is fictitious.

Analogy: A Ball on a String

The string pulls inward.

Imagine swinging a ball on a string in a horizontal circle. The **tension** in the string provides the centripetal force, pulling the ball toward your hand (the center). If the string breaks, the ball flies off in a straight line tangent to the circle — not outward — because no force is acting on it.

Takeaway. The string tension is a physical example of centripetal force.

Problem-Solving Workflow

A step-by-step approach.

Takeaway. Follow these steps to solve centripetal acceleration problems systematically.

Worked Example: 5 m/s at 2 m radius

A quick numerical check.

A ball is tied to a string and swung in a horizontal circle of radius 2 m with a constant speed of 5 m/s. Calculate the centripetal acceleration.

Takeaway. For v=5 m/s and r=2 m, the centripetal acceleration is 12.5 m/s².

Warning: Not a Force Itself

Acceleration is not a force.

Centripetal **acceleration** is not a force. It is the acceleration that results from a net force pointing toward the center. The force that causes this acceleration (tension, gravity, friction, etc.) is called the **centripetal force**.

Takeaway. Centripetal acceleration is a result, not a force itself.