EQUILIBRIUM

ROTATIONAL KINETIC ENERGY

ANGULAR MOMENTUM

EquilibriumTwo Conditions for Equilibrium

1) Sum of Forces must be zero

Tacos ! (m= 10 kg)

Determine the Force on Each Chain

EquilibriumTwo Conditions for Equilibrium

2) Sum of Torques must be zero

Tacos ! (m=5kg)

Determine the Tension on the Chain

T=?

Length 1.50m

50o

X

mg

T

EquilibriumTwo Conditions for Equilibrium

2) Sum of Torques must be zero

m= 4.0 kg

Pizza! (m=2.0kg)

2.0m1.3 m45o

Determine the Tension on the Chain X

mg

T

mg

EquilibriumTwo Conditions for Equilibrium

1) Sum of Forces must be zero2) Sum Torques must be zero

Length 3.0m

Tacos ! (m= 10 kg)

X=0m X=2.0m

Determine the Force on Each Chain

mg

T1 T2

X

EquilibriumTwo Conditions for Equilibrium

1) Sum of Forces must be zero2) Sum Torques must be zero

L= 5.0,, m=15kg

m= 50kg, x=3.0m

F1 =? F2 =?

mg

T1 T2

X

mg

Energy in Rotational MotionObjects in motion have Kinetic Energy

Rotating Objects have Kinetic Energy

Energy of single particle:

𝐾𝐸 =1

2𝑚𝑣2 → 𝑣 = 𝜔𝑟 →

1

2𝑚 𝜔𝑟 2

𝐾𝐸 =1

2𝑚𝑟2 𝜔2 → 𝑚𝑟2 = 𝐼 = 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝐼𝑛𝑒𝑟𝑡𝑖𝑎 =

𝑖

𝑚𝑖𝑟𝑖2

𝐾𝐸 =1

2𝐼𝜔2

Energy in Rotational Motion

A force of 10N acts for 2.0m on 20 kg Solid Disc with a radius of 0.2m. What is the final angular speed of the wheel as it spins on its axis?

F = 10Nd=2m

m=20kgr=0.20m

𝑊 = ∆𝐾𝐸

𝐼 =1

2𝑚𝑟2

+

Energy in Rotational Motion• Rolling Objects have both Translational Kinetic Energy and Rotational

Kinetic Energy

• Total Energy =KEt+KEr = 1

2𝑚𝑣2+

1

2𝐼𝜔2

• Find the total energy of a 4.0kg bowling ball moving at 8 m/s. (𝐼 =2

5𝑚𝑟2)

Conservation of Energy

m=1.5 kgr=0.20m

Basketball

1.5m

V=?

GPE

KE + RKE

𝐺𝑃𝐸 = 𝐾𝐸𝑇 + 𝐾𝐸𝑅 → simplify

Gravitational Potential Converted to both translational and rotational kinetic energy.

Conservation of Energy

• Hollow Sphere

• Hoop

• Disc,

• Solid Sphere

Which is which?

How can you tell?

Conservation of Energy• Which has more energy? Hoop, disc? (m=2kg, r =0.40m, vi=6m/s)

• How high up the ramp will each go?

Angular Momentum• Momentum for a point mass rotating around an axis.

• 𝐿 = 𝑟 × 𝑚𝑣 → 𝑣 = 𝜔𝑟 → 𝐿 = 𝑟 ×𝑚 𝜔𝑟 = 𝑚𝑟2𝜔 → 𝑚𝑟2 = 𝐼

• 𝐿 = 𝑚𝑟2𝜔

•𝐿 = 𝐼𝜔• 𝑝 = 𝑚𝑣 ↔ 𝐿 = 𝐼𝜔

Right Hand Rule

Angular Momentum

• Determine the angular momentum of a 2.0kg bike wheel, spinning at 200rpm. (r=0.35m)

Determine the angular momentum of a 5.0kg dog, on the outside edge of a merry go round, moving at 2.0 m/s at 1.2m from the center.

Conservation of Angular Momentum• 𝐿1 = 𝐿2 → 𝐼𝜔 = 𝐼𝜔

Conservation of Angular Momentum

𝐿1 = 𝐿2 → 𝐼𝜔 = 𝐼𝜔

A figure skater with an Initial Moment of Inertia of 3.8 kg*m2 isspinning at 30 rpm, stands and accelerates to 90 rpm. What is her new moment of Inertia?

Conservation of Angular Momentum• 𝐿1 = 𝐿2 → 𝐼𝜔 = 𝐼𝜔

• 𝑚𝑣1𝑟1 = 𝑚𝑣2𝑟2

Mass of Blocks = 2kgR1=0.4mV1 = 5m/s

R2 = 0.15V2=?

Conservation of Angular Momentum

M=0.80kgL=0.60m

M=1.2kgr=0.30mω1=9.0rad/sω2=?

𝐿1 = 𝐿2 → 𝐼𝜔 + 𝐼𝜔 = 𝐼𝜔 + 𝐼𝜔

A B