 #1
hidemi
 208
 36
 Homework Statement:

A meter stick is pivoted at a point a distance a from its center and swings as a physical pendulum. Of the following values for a, which results in the shortest period of oscillation?
A. a = 0.1m
B. a = 0.2m
C. a = 0.3m
D. a = 0.4m
E. a = 0.5m
The answer is C.
 Relevant Equations:
 T = 2 π √(I/k)
mg*sinθ*d =(1/12 * m^2* 1^2 + ma^2) θ"
(since θ is very very small, sin θ = θ)
θ" + [ga/ (1/12 + a^2)] θ =0
T (period) = 2π / ω
ω (0.1m) = 9.8 * 0.1 / (1/12+0.1^1) = 10.5
ω (0.2m) = 9.8 * 0.2 / (1/12+0.2^1) = 15.9
ω (0.3m) = 9.8 * 0.3 / (1/12+0.3^1) = 17.0
ω (0.4m) = 9.8 * 0.4 / (1/12+0.4^1) = 16.1
ω (0.5m) = 9.8 * 0.5 / (1/12+0.5^1) = 14.7
Therefore, C is the correct answer. However, is there a way to explain this phenomenon of why a=0.3m hits the minimum?
(since θ is very very small, sin θ = θ)
θ" + [ga/ (1/12 + a^2)] θ =0
T (period) = 2π / ω
ω (0.1m) = 9.8 * 0.1 / (1/12+0.1^1) = 10.5
ω (0.2m) = 9.8 * 0.2 / (1/12+0.2^1) = 15.9
ω (0.3m) = 9.8 * 0.3 / (1/12+0.3^1) = 17.0
ω (0.4m) = 9.8 * 0.4 / (1/12+0.4^1) = 16.1
ω (0.5m) = 9.8 * 0.5 / (1/12+0.5^1) = 14.7
Therefore, C is the correct answer. However, is there a way to explain this phenomenon of why a=0.3m hits the minimum?