Classical Mechanics - Angular Speed

A thin plate of mass M, length L, and width 2d is mounted vertically on a frictionless axle along the z-axis. Initially the object is at rest. It is then tapped with a hammer to provide a torque τ, which produces an angular impulse H about the z-axis of magnitude H = ∫ τ dt. What is the angular speed ω of the plate about the z-axis after the tap? 

A. H/2Md²
B. H/Md²
C. 2H/Md²
D. 3H/Md²
E. 4H/Md²

(GR9277 #82)

Solution:

H = ∫ τ dt

Torque: τ = Iα
Angular acceleration: α = ω/t → ω = αt

H = ∫ τ dt = Iα ∫dt = Iαt = Iω

Moment inertia for the plate about the z-axis: I = ¹/₃Md²

H = ¹/₃Md²ω
ω = 3H/Md²

Answer: D

Classical Mechanics - Impulse

The figure shows a plot of the time dependent force F_x(t) acting on a particle in motion along the x-axis. What is the total impulse delivered to the particle?

A. 0
B. 1 kg m/s
C. 2 kg m/s
D. 3 kg m/s
E. 4 kg m/s

(GR0177 #54)

Solution:

Impulse: I = ∆p = F ∆t
Area =  ½(2)(2) = 2 kg m/s

Answer: C