PHY231 Review Session: Momentum, Angular Displacement, and Gravitation, Study notes of Physics

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• Review Session for Midterm
  • Chapter 6, 7, 8 and

Momentum - impluse

• Momentum - SI unit kg.m/s p =ˆ mv F = ma = Δ p Δ t
• Newton revisited
• Impulse Δ p = F * Δ t

Elastic/inelastic collisions

• Two main types of collisions
  • Inelastic
  • Elastic
• Total momentum is conserved for both elastic and inelastic collisions
• Total kinetic energy is also conserved for elastic collisions

Summary - collisions

• Total momentum conserved for an isolated system
• Collisions of two objects (1D)
  • Perfectly inelastic
    • If m 1 =m 2
  • Elastic (KE i =KE f )
    • If m 1 =m 2 PHY231 5

p

total , f

= p

total , i

m

1

v

1 f

+ m

2

v

2 f

= m

1

v

1 i

+ m

2

v

2 i

m

1

+ m

( 2 )

v

f

= m

1

v

1 i

+ m

2

v

2 i

m

1

v

1 f

+ m

2

v

2 f

= m

1

v

1 i

+ m

2

v

2 i

v

1 i

+ v

1 f

= v

2 i

+ v

2 f

2 v

f

= v

1 i

+ v

2 i

v

2 f

= v

1 i

v

1 f

= v

2 i

Angular velocity

• Average angular velocity
• SI unit: rad/s
• The angular velocity tells you how fast an object is rotating. It is sometimes expressed in rpm (rotation per minute) - 1 rpm = 2*π/60 = 0.1047…rad/s ω =ˆ Δθ Δ t = θ f − θ i Δ t

Tangential angular acceleration

• Average tangential angular acceleration SI unit: rad/s 2
• The angular acceleration tells you how much the angular velocity is changing through time.
• α is related to a t tangential acceleration a t = α*r
• Do not confuse them with a c centripetal acceleration α =ˆ Δω Δ t = ω f − ω i Δ t

Tangential velocity and angular velocity

• tangential velocity and angular velocity are not the same thing
• An object moves on a circle of radius r=2.0 m at an angular velocity ω=10 rad/s. Its velocity is

v =

Δ s

Δ t

r Δ θ

Δ t

= r ω

v = r ω v = r ω = 2.0 * 10 = 20 m / s

Rotation of an object

• For a solid object rotating
  • All the points of the object have equal angular velocity
  • The points farther away from the axis of rotation have greater tangential velocity ω same for all points v = ω r depends on r

Constant acceleration

• There is a direct parallel between linear motion and rotational motion a = constant Δ v = a Δ t Δ x = v i Δ t + 1 2 a Δ t 2 v f 2 = v i 2

• 2 a Δ x α = constant Δ ω = αΔ t Δ θ = ω i Δ t + 1 2 αΔ t 2 ω f 2 = ω i 2
• 2 αΔ θ

Gravitation (general formulation)

• Near Earth’s surface F g can be approximated by F

g

= mg g ≈ 9.81 m / s 2

Escape speed

• We apply conservation of energy
• After some algebra we find
• Calculating for Earth we would find v esc ~11.2 km/s € KE i

• GPE i = KE f
• GPE f € v esc = 2 GM R

Kepler’s laws

• First Law
  • Consequence of force proportional to 1/r 2
• Second Law
  • Area SAB =Area SCD if
  • Δt AB =Δt CD

Extended objects

• Extended objects have two motions
  • Translational motion
    • F=ma with F the net force acting on the object and a the acceleration of the object’s center of mass
  • Rotational motion
    • τ=Iα

Center of mass (or gravity)

• The (2-D) position of the object’s center of mass is defined as
• With
• It is different than the object’s geometric center
• If you hang the object at r CM it doesn’t move

rCM = x CM y CM

€ x CM = m j x j j ∑ m j j ∑ € y CM = m j y j j ∑ m j j ∑