CIE AS & A Level Physics 9702: Topic 12: Motion in a circle- Unit : 12.1 Kinematics of uniform circular motion Study Notes

                                                                          Radians & Angular Displacement

•  In circular motion, it is more convenient to measure angular displacement in units of radians rather than units of degrees
•  The angular displacement $(\theta)$ of a body in circular motion is defined as:
• The change in angle, in radians, of a body as it rotates around a circle
• The angular displacement is the ratio of:

$
\Delta \theta=\frac{\text { distance travelled around the circle }}{\text { radius of the circle }}
$

• Note: both distances must be measured in the same units e.g. metres
• A radian $(r a d)$ is defined as:
• The angle subtended at the centre of a circle by an arc equal in length to the radius of the circle
•  Angular displacement can be calculated using the equation:

$
\Delta \theta=\frac{\Delta s}{r}
$

When the angle is equal to one radian, the length of the arc (Δs) is equal to the radius (r) of the circle

Where:

• Δθ = angular displacement, or angle of rotation (radians)
• s = length of the arc, or the distance travelled around the circle (m)
• r = radius of the circle (m)

•  Radians are commonly written in terms of $\pi$
• The angle in radians for a complete circle $\left(360^{\circ}\right)$ is equal to:

$
\frac{\text { circumference of circle }}{\text { radius }}=\frac{2 \pi r}{r}=2 \pi
$

• If an angle of $360^{\circ}=2 \pi$ radians, then 1 radian in degrees is equal to:

$
\frac{360}{2 \pi}=\frac{180}{\pi} \simeq 57.30^{\circ}
$

• Use the following equation to convert from degrees to radians:

$
\theta^{\circ} \times\left(\frac{\pi}{180}\right)=\theta \text { rad }
$
                                    Table of common degrees to radians conversions

Exam Tip

•  You will notice your calculator has a degree (Deg) and radians (Rad) mode
•  This is shown by the “D” or ” $R$ ” highlighted at the top of the screen
•  Remember to make sure it’s in the right mode when using trigonometric functions (sin, cos, tan) depending on whether the answer is required in degrees or radians
• It is extremely common for students to get the wrong answer (and lose marks) because their calculator is in the wrong mode – make sure this doesn’t happen to you!