Rotational motion
Core concepts
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑠
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 = 𝑟𝑎𝑑𝑖𝑢𝑠
(θ = 𝑟
)
𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 ∆θ
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑡𝑖𝑚𝑒
(ω = ∆𝑡
)
2π 2π
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 2π × 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
(ω = 2π𝑓 = 𝑇
)
𝑙𝑖𝑛𝑒𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑣
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑟𝑎𝑑𝑖𝑢𝑠
(ω = 𝑟 )
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 ∆ω
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑡𝑖𝑚𝑒
(α = ∆𝑡 )
𝑙𝑖𝑛𝑒𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑎
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑟𝑎𝑑𝑖𝑢𝑠
(α = 𝑟 )
Angular motion
● ω2 = ω1 + α𝑡
2 2
● ω2 = ω1 + 2αθ
2
α𝑡
● θ = ω1𝑡 + 2
2
α𝑡
● θ = ω2𝑡 − 2
ω1+ω2
● θ = 2
𝑡
Graphs
● Displacement-time: gradient is angular velocity
● Velocity-time: gradient is acceleration, area is displacement (radians)
● Acceleration-time: area is angular velocity
Torque
● Torque must be applied to make rotating object spin faster
● 𝑡𝑜𝑟𝑞𝑢𝑒 = 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑓𝑜𝑟𝑐𝑒 × 𝑟𝑎𝑑𝑖𝑢𝑠 (𝑇 = 𝐹𝑟)
Inertia
● Inertia: an object’s resistance to change in motion
● Rotational inertia: an object’s resistance to change in rotational motion
● Newton’s first law is law of inertia: an object will remain at rest or continue to move with
constant velocity unless acted on by a resultant force
2 2
● 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 = 𝑚𝑎𝑠𝑠 × 𝑟𝑎𝑑𝑖𝑢𝑠 (𝐼 = 𝑚𝑟 ) for a point mass
2 2
● 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 = Σ(𝑚𝑎𝑠𝑠 × 𝑟𝑎𝑑𝑖𝑢𝑠 ) (𝐼 = Σ𝑚𝑟 ) for any object
● Inertia and torque:
○ From Newton’s second law: Σ𝐹 = 𝑚𝑎
○ 𝐹𝑟 = 𝑚𝑟𝑎
2
○ 𝑎 = 𝑟α so 𝐹𝑟 = 𝑚𝑟 α
2
○ 𝐼 = 𝑚𝑟 so 𝑡𝑜𝑟𝑞𝑢𝑒 = 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 × 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 (𝑇 = 𝐼α)
● Rotational/torque form of Newton’s second law: a resultant torque causes angular
acceleration. Torque causes a change in rotational motion
Energy
1 2 1 2 2 1 2
● Derivation: 𝐸𝑘 = 2
𝑚𝑣 , 𝑣 = ω𝑟 so 𝐸𝑘 = 2
𝑚𝑟 ω = 2
𝐼ω
1 2 1 2
● 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 = 2
𝑖𝑛𝑒𝑟𝑡𝑖𝑎 × 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝐸𝑘 = 2
𝐼ω )
● Total KE of an object will be the sum of its rotational and linear (translational) KE
● Alternate derivation: 𝑊 = 𝐹𝑑 = 𝐹𝑟θ = 𝑇θ
, ● 𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝑡𝑜𝑟𝑞𝑢𝑒 × 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 (𝑊 = 𝑇θ)
Power
𝑊 θ
● Derivation: 𝑃 = 𝑡
=𝑇 𝑡
= 𝑇ω
● 𝑝𝑜𝑤𝑒𝑟 = 𝑡𝑜𝑟𝑞𝑢𝑒 × 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝑃 = 𝑇ω)
Flywheels
● Large heavy wheels which have lots of inertia
● Used to smooth out torque and angular speed and can store energy
● Angular speed of flywheels limited by breaking stress of material - flywheels from new
composites can now spin up to over 50,000rpm
● Engines:
○ Helps steady position of shaft when fluctuating torque exerted on it by
pistons - smooths out pulsing from engine
○ Crankshaft translates linear oscillations of piston into rotational to
drive wheels
○ Flywheel stores energy and releases it when engine isn’t delivering power - helps
ensure wheels turn smoothly and help wheels turn when engine is idling
● KERS (Kinetic Energy Recovery System):
○ Recovers moving vehicle’s energy during braking - stored in reservoir (flywheel or
high voltage batteries) for use when accelerating
○ Adds more power while increasing efficiency
○ Used in F1 cars and now on some London buses
● Machine tools:
○ Heavy machinery requires lot of work done in very short amount of time
○ Machines to work sheet metal or punching out parts include flywheel to assist motor
and supply energy quickly - motor would stall on its own as requirement spike too big
Measuring inertia of flywheels
● Use falling mass and record time taken for mass to spin flywheel and fall a
measured height - used to calculate GPE loss of mass 𝐺𝑃𝐸𝑚𝑎𝑠𝑠 = 𝑚𝑔ℎ
● Use radius of axle around which string is wrapped to find number of rotations
ℎ 2π𝑛 ℎ
𝑛= 2π𝑟
- ω𝑎𝑣𝑒 = 𝑡
= 𝑟𝑡
● Assuming constant acceleration, ω𝑓𝑖𝑛𝑎𝑙 = 2ω𝑎𝑣𝑒
2 2 2
𝑚𝑣 𝑚ω 𝑟
● To find velocity of mass 𝑣 = ω𝑟 - 𝐾𝐸𝑚𝑎𝑠𝑠 = 2
= 2
● 𝐾𝐸𝑓𝑙𝑦𝑤ℎ𝑒𝑒𝑙 = 𝐺𝑃𝐸𝑚𝑎𝑠𝑠 − 𝐾𝐸𝑚𝑎𝑠𝑠 - measure radius of flywheel to find inertia and mass
1 2 1 2 2
𝐾𝐸𝑓𝑙𝑦𝑤ℎ𝑒𝑒𝑙 = 2
𝐼ω = 2
𝑚𝑟 ω
● Actual inertia likely less due to friction between bearings and axle, air resistance
Momentum
2
● Derivation: 𝐿 = 𝑝𝑟 = 𝑚𝑣𝑟, 𝑣 = ω𝑟 so 𝐿 = 𝑚𝑟 ω = 𝐼ω
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 × 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝐿 = 𝐼ω)
● Law of conservation of angular momentum: the angular momentum about an axis is constant
if no external torque acts about the axis
● Impulse: as per Newton’s second law 𝑓𝑜𝑟𝑐𝑒 × 𝑡𝑖𝑚𝑒 = ∆𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 (𝐹𝑡 = ∆𝑚𝑣),
replacing for angular version means 𝑡𝑜𝑟𝑞𝑢𝑒 × 𝑡𝑖𝑚𝑒 = ∆𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 (𝑇𝑡 = ∆𝐼ω)
● Means as inertia decreases, angular velocity increases
○ If a person is spinning with weights pulls weights towards the centre
then they spin faster
○ When a diver jumps off the board and tucks their radius decreases,
decreasing inertia and so increasing speed of roll
Core concepts
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑠
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 = 𝑟𝑎𝑑𝑖𝑢𝑠
(θ = 𝑟
)
𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 ∆θ
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑡𝑖𝑚𝑒
(ω = ∆𝑡
)
2π 2π
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 2π × 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
(ω = 2π𝑓 = 𝑇
)
𝑙𝑖𝑛𝑒𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑣
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑟𝑎𝑑𝑖𝑢𝑠
(ω = 𝑟 )
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 ∆ω
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑡𝑖𝑚𝑒
(α = ∆𝑡 )
𝑙𝑖𝑛𝑒𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑎
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑟𝑎𝑑𝑖𝑢𝑠
(α = 𝑟 )
Angular motion
● ω2 = ω1 + α𝑡
2 2
● ω2 = ω1 + 2αθ
2
α𝑡
● θ = ω1𝑡 + 2
2
α𝑡
● θ = ω2𝑡 − 2
ω1+ω2
● θ = 2
𝑡
Graphs
● Displacement-time: gradient is angular velocity
● Velocity-time: gradient is acceleration, area is displacement (radians)
● Acceleration-time: area is angular velocity
Torque
● Torque must be applied to make rotating object spin faster
● 𝑡𝑜𝑟𝑞𝑢𝑒 = 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑓𝑜𝑟𝑐𝑒 × 𝑟𝑎𝑑𝑖𝑢𝑠 (𝑇 = 𝐹𝑟)
Inertia
● Inertia: an object’s resistance to change in motion
● Rotational inertia: an object’s resistance to change in rotational motion
● Newton’s first law is law of inertia: an object will remain at rest or continue to move with
constant velocity unless acted on by a resultant force
2 2
● 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 = 𝑚𝑎𝑠𝑠 × 𝑟𝑎𝑑𝑖𝑢𝑠 (𝐼 = 𝑚𝑟 ) for a point mass
2 2
● 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 = Σ(𝑚𝑎𝑠𝑠 × 𝑟𝑎𝑑𝑖𝑢𝑠 ) (𝐼 = Σ𝑚𝑟 ) for any object
● Inertia and torque:
○ From Newton’s second law: Σ𝐹 = 𝑚𝑎
○ 𝐹𝑟 = 𝑚𝑟𝑎
2
○ 𝑎 = 𝑟α so 𝐹𝑟 = 𝑚𝑟 α
2
○ 𝐼 = 𝑚𝑟 so 𝑡𝑜𝑟𝑞𝑢𝑒 = 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 × 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 (𝑇 = 𝐼α)
● Rotational/torque form of Newton’s second law: a resultant torque causes angular
acceleration. Torque causes a change in rotational motion
Energy
1 2 1 2 2 1 2
● Derivation: 𝐸𝑘 = 2
𝑚𝑣 , 𝑣 = ω𝑟 so 𝐸𝑘 = 2
𝑚𝑟 ω = 2
𝐼ω
1 2 1 2
● 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 = 2
𝑖𝑛𝑒𝑟𝑡𝑖𝑎 × 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝐸𝑘 = 2
𝐼ω )
● Total KE of an object will be the sum of its rotational and linear (translational) KE
● Alternate derivation: 𝑊 = 𝐹𝑑 = 𝐹𝑟θ = 𝑇θ
, ● 𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝑡𝑜𝑟𝑞𝑢𝑒 × 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 (𝑊 = 𝑇θ)
Power
𝑊 θ
● Derivation: 𝑃 = 𝑡
=𝑇 𝑡
= 𝑇ω
● 𝑝𝑜𝑤𝑒𝑟 = 𝑡𝑜𝑟𝑞𝑢𝑒 × 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝑃 = 𝑇ω)
Flywheels
● Large heavy wheels which have lots of inertia
● Used to smooth out torque and angular speed and can store energy
● Angular speed of flywheels limited by breaking stress of material - flywheels from new
composites can now spin up to over 50,000rpm
● Engines:
○ Helps steady position of shaft when fluctuating torque exerted on it by
pistons - smooths out pulsing from engine
○ Crankshaft translates linear oscillations of piston into rotational to
drive wheels
○ Flywheel stores energy and releases it when engine isn’t delivering power - helps
ensure wheels turn smoothly and help wheels turn when engine is idling
● KERS (Kinetic Energy Recovery System):
○ Recovers moving vehicle’s energy during braking - stored in reservoir (flywheel or
high voltage batteries) for use when accelerating
○ Adds more power while increasing efficiency
○ Used in F1 cars and now on some London buses
● Machine tools:
○ Heavy machinery requires lot of work done in very short amount of time
○ Machines to work sheet metal or punching out parts include flywheel to assist motor
and supply energy quickly - motor would stall on its own as requirement spike too big
Measuring inertia of flywheels
● Use falling mass and record time taken for mass to spin flywheel and fall a
measured height - used to calculate GPE loss of mass 𝐺𝑃𝐸𝑚𝑎𝑠𝑠 = 𝑚𝑔ℎ
● Use radius of axle around which string is wrapped to find number of rotations
ℎ 2π𝑛 ℎ
𝑛= 2π𝑟
- ω𝑎𝑣𝑒 = 𝑡
= 𝑟𝑡
● Assuming constant acceleration, ω𝑓𝑖𝑛𝑎𝑙 = 2ω𝑎𝑣𝑒
2 2 2
𝑚𝑣 𝑚ω 𝑟
● To find velocity of mass 𝑣 = ω𝑟 - 𝐾𝐸𝑚𝑎𝑠𝑠 = 2
= 2
● 𝐾𝐸𝑓𝑙𝑦𝑤ℎ𝑒𝑒𝑙 = 𝐺𝑃𝐸𝑚𝑎𝑠𝑠 − 𝐾𝐸𝑚𝑎𝑠𝑠 - measure radius of flywheel to find inertia and mass
1 2 1 2 2
𝐾𝐸𝑓𝑙𝑦𝑤ℎ𝑒𝑒𝑙 = 2
𝐼ω = 2
𝑚𝑟 ω
● Actual inertia likely less due to friction between bearings and axle, air resistance
Momentum
2
● Derivation: 𝐿 = 𝑝𝑟 = 𝑚𝑣𝑟, 𝑣 = ω𝑟 so 𝐿 = 𝑚𝑟 ω = 𝐼ω
● 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 × 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 (𝐿 = 𝐼ω)
● Law of conservation of angular momentum: the angular momentum about an axis is constant
if no external torque acts about the axis
● Impulse: as per Newton’s second law 𝑓𝑜𝑟𝑐𝑒 × 𝑡𝑖𝑚𝑒 = ∆𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 (𝐹𝑡 = ∆𝑚𝑣),
replacing for angular version means 𝑡𝑜𝑟𝑞𝑢𝑒 × 𝑡𝑖𝑚𝑒 = ∆𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 (𝑇𝑡 = ∆𝐼ω)
● Means as inertia decreases, angular velocity increases
○ If a person is spinning with weights pulls weights towards the centre
then they spin faster
○ When a diver jumps off the board and tucks their radius decreases,
decreasing inertia and so increasing speed of roll
