## 06. Motion in a Circle

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[accordion title="1. Rotational Kinematics"]

• Angular displacement $\theta$ is defined as the angle an object turns with respect to the centre of a circle. $\theta=\dfrac{s}{r}$ where s is the arc and r is the radius of the circle.
• One radian is the angular displacement when the arc length is equal to the radius of the circle.
• Angular velocity $\omega$ is defined as the rate of change of angular displacement. $\omega=\dfrac{d\theta}{dt}$
• For motion in a circle of fixed radius, $\omega=\dfrac{d\theta}{dt}=\dfrac{d(\dfrac{s}{r})}{dt}=\dfrac{1}{r}\dfrac{ds}{dt}=\dfrac{v}{r}$. Thus $v=r\omega$.
• Average angular velocity in one cycle. $\omega=\dfrac{2\pi}{T}=2\pi f$ where T is the period and f is the frequency.

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[accordion title="2. Centripetal Force"]

• Centripetal acceleration $a=\dfrac{v^2}{r} = r\omega^2$.
• Centripetal force $F =\dfrac{mv^2}{r} = mr\omega^2$.

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[accordion title="3. Uniform Circular Motion"]

For a body in uniform circular motion, there is a change in velocity as the direction of motion is changing. This requires an acceleration that is perpendicular to the velocity and directed towards the centre of circle. This acceleration is provided by a centripetal force. A resultant force acting on a body toward the centre of a circle provides the centripetal force.

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[accordion title="4. Non-Uniform Circular Motion"]

Learn how to solve problems on circular motions of conical pendulum, cyclist, car, aircraft, swinging a pail etc.

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## Cardboard Boomerang

A indoor boomerang can be constructed using 3 strips of cardboard put together. Throwing it may require some practice though but when you get the hang of it, it can inject great fun into your lesson. You can explore using different types of material to get the best boomerang.

Materials

1. Cardboard about 1 mm thick, of suitable rigidity
2. Staples
3. Scissors
4. Rubber band or tape for added weight

Procedure

1. Cut 3 equal rectangular strips of cardboard measuring 12 cm x 2.5 cm. You may like to trim the sharp corners on one of the ends of each strip.
2. Cut a slit of 1.5 cm along the middle of each strip, on the untrimmed end.
3. Join the strips together at the slits, the angle between two adjacent strips being 120 degrees.

4. One side of the slit should overlap another so that it looks like the above:
5. Staple the overlapping centre together.
6. The boomerang is ready for use! Throwing the boomerang is done by holding onto one of the wings. The boomerang should be almost vertical, at an angle of about 10o. With a flick of the wrist, spin the boomerang as it leaves the hand. The direction of spin should be toward the side that is tilted up.

Science Explained
A boomerang requires a centripetal force to cause it to fly in a circular path back to the thrower. This centripetal force comes from the lift that the wings generate as they cut through the air.