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# Rotational motion

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• 1. Rotational Motion<br />“I’m spinning around, <br />move outta my way…”<br />
• 2. Angular Displacement:<br />We use radians for rotational motion.<br />𝜃 is the angular displacement, “what angle has the object rotated through?”<br /> <br />𝜋4<br /> <br />3𝜋4<br /> <br />2𝜋<br /> <br />d<br />𝜋<br /> <br />
• 3. What is a radian?<br />1 radian is about a sixth of a whole circle<br />(about 57°)<br />So…<br />There are just over 6 radians in a circle.<br />
• 4. Angular Velocity:<br />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)<br />𝛼 is the angular acceleration,  𝜶=∆𝝎∆𝒕  𝒓𝒂𝒅𝒔−𝟐<br /> <br />Other useful <br />formulae:<br />𝝎=𝟐𝝅𝑻<br />𝝎=𝟐𝝅𝒇<br />𝒘𝒉𝒆𝒓𝒆 𝑻=𝒑𝒆𝒓𝒊𝒐𝒅,<br />𝒇=𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚.<br /> <br />𝑂𝑛𝑒 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑓𝑜𝑟  𝑜𝑛𝑒 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛<br /> <br />𝑓=1𝑇<br /> <br />
• 5. Finding Linear Quantities:<br />Circumfrence = 2𝜋𝑟<br />So the distance will be:<br />𝑑2𝜋𝑟=𝜃2𝜋<br />𝑑𝑟=𝜃<br />𝑑=𝑟𝜃<br />Similarly for 𝜔 and 𝛼<br /> <br />ANGULAR<br />d<br />𝒅=𝒓𝜽<br />𝒗=𝒓𝝎<br />𝒂=𝒓𝜶<br /> <br />LINEAR<br />
• 6. Kinematics<br />All our old mates the familiar kinematics for linear motion still hold for rotational motion although with new symbols.<br />𝜔𝑓=𝜔𝑖+𝛼𝑡<br />𝜔𝑓2=𝜔𝑓2 +2𝛼𝜃<br />𝜃=𝜔𝑖𝑡+12𝛼𝑡2<br /> <br />
• 7. Torque and Angular Acceleration<br /> Linear force -> Linear acceleration<br />Angular force (torque) -> Angular acceleration<br />𝝉=𝑭𝒓 (measured in Nm)<br />Angular acceleration is proportional to the applied force.<br />𝝉=𝑰𝜶 <br />𝐼 is the constant of proportion, it is the rotational inertia.<br />Rotational inertia depends on the shape of the object (but we will get into that).<br /> <br />
• 8. Rotational Inertia<br />Mass has the property of inertia, more mass is harder to move (even in when there is no gravity).<br />An object with a large rotational inertia is hard to start spinning (even in when there is no gravity).<br />Rotational inertia is not dependent weight, it is dependent on the distribution of the mass as well as the amount of mass.<br /> 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.<br />
• 9. Changing your Rotational Inertia:<br />
• 10. Stability and Rotational Inertia:<br />The more rotational inertia an object has the more stable it is.<br />Because it is harder to move ∴ it must be harder to destabilise. <br />The stability of an object depends on the torques produced by its weight.<br />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.<br />So a bike wheel is more stable than a disk of the same mass. <br />The faster an object rotates the more stable it will become. <br />(but we will get into that later)<br /> <br />
• 11. Calculating Rotational Inertia:<br />𝑰=𝒎𝒓𝟐<br />For a dumbbell: 𝐼=𝑚𝑟2=𝑚1𝑟12+𝑚2𝑟22<br />Only two masses 𝑚1 and 𝑚2.<br />For a hoop: 𝐼=𝑚𝑟2=𝑚𝑟2<br />All the masses are the same distance, r, <br />from the centre<br /> <br />COM<br />𝒎𝟐<br /> <br />𝒎𝟏<br /> <br />𝑟2<br /> <br />𝑟1<br /> <br />𝑟<br /> <br />