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