Rotational Motion“I’m spinning around,

move outta my way…”

Angular Displacement:• We use radians for rotational motion.• is the angular displacement, “what angle has the object rotated

through?”

2𝜋𝜋𝜋43𝜋

4d

What is a radian?

1 radian is about a sixth of a whole circle(about 57°)So…There are just over 6 radians in a circle.

Angular Velocity:• is the angular velocity, the angle the object has turned through

divided by the time taken to do it,

• is the angular acceleration Other useful formulae:

𝑂𝑛𝑒𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑡𝑖𝑚𝑒𝑡𝑎𝑘𝑒𝑛 𝑓𝑜𝑟 𝑜𝑛𝑒𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛

𝑓 =1𝑇

Finding Linear Quantities:

ANGULARLINEAR

d

Circumfrence = So the distance will be:

Similarly for and

Kinematics• All our old mates the familiar kinematics for linear motion still

hold for rotational motion although with new symbols.

Torque and Angular Acceleration

Linear force → Linear accelerationAngular force (torque) → Angular acceleration

» (measured in Nm)• Angular acceleration is proportional to the

applied force.»

• is the constant of proportion, it is the rotational inertia.

Rotational inertia depends on the shape of the object (but we will get into that).

Rotational Inertia• Mass has the property of inertia, more mass is harder to move

(even in when there is no gravity).• An object with a large rotational inertia is hard to start spinning

(even in when there is no gravity).

• Rotational inertia is not dependent weight, it is dependent on the distribution of the mass as well as the amount of mass.

• An object with most of it’s mass further from the centre (rotational axis) will have a large rotational inertia e.g. a bike wheel.

Changing your Rotational Inertia:

Stability and Rotational Inertia:

• The more rotational inertia an object has the more stable it is.

Because it is harder to move it must be harder to destabilise.

• The stability of an object depends on the torques produced by its weight.

i.e. the further the masses are from the COM the larger the torque they produce and therefore more force is required to destabilise it.So a bike wheel is more stable than a disk of the same mass.

• The faster an object rotates the more stable it will become.

(but we will get into that later)

Calculating Rotational Inertia:

For a dumbbell: Only two masses and

For a hoop: All the masses are the same distance, r, from the centre

𝒎𝟏 𝒎𝟐

COM

𝑟1 𝑟2

𝑟