Rotational Motion “I’m spinning around, move outta my way…”
Angular Displacement: We use radians for rotational motion. 𝜃 is the angular displacement, “what angle has the object rotated through?”
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: 𝜔 is the angular velocity, the angle the object has turned through divided by the time taken to do it, 𝝎=∆𝜽∆𝒕 (𝒓𝒂𝒅𝒔−𝟏) 𝛼 is the angular acceleration, 𝜶=∆𝝎∆𝒕 𝒓𝒂𝒅𝒔−𝟐
Other useful formulae: 𝝎=𝟐𝝅𝑻 𝝎=𝟐𝝅𝒇 𝒘𝒉𝒆𝒓𝒆 𝑻=𝒑𝒆𝒓𝒊𝒐𝒅, 𝒇=𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚.
𝑂𝑛𝑒 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑓𝑜𝑟 𝑜𝑛𝑒 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛
Finding Linear Quantities: Circumfrence = 2𝜋𝑟 So the distance will be: 𝑑2𝜋𝑟=𝜃2𝜋 𝑑𝑟=𝜃 𝑑=𝑟𝜃 Similarly for 𝜔 and 𝛼
ANGULAR d 𝒅=𝒓𝜽 𝒗=𝒓𝝎 𝒂=𝒓𝜶
Kinematics All our old mates the familiar kinematics for linear motion still hold for rotational motion although with new symbols. 𝜔𝑓=𝜔𝑖+𝛼𝑡 𝜔𝑓2=𝜔𝑓2 +2𝛼𝜃 𝜃=𝜔𝑖𝑡+12𝛼𝑡2
Torque and Angular Acceleration Linear force -> Linear acceleration Angular force (torque) -> Angular acceleration 𝝉=𝑭𝒓 (measured in Nm) Angular acceleration is proportional to the applied force. 𝝉=𝑰𝜶 𝐼 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: 𝑰=𝒎𝒓𝟐 For a dumbbell: 𝐼=𝑚𝑟2=𝑚1𝑟12+𝑚2𝑟22 Only two masses 𝑚1 and 𝑚2. For a hoop: 𝐼=𝑚𝑟2=𝑚𝑟2 All the masses are the same distance, r, from the centre