Consider a simple pendulum. Does the angular acceleration of the mass remain con
ID: 1474628 • Letter: C
Question
Consider a simple pendulum. Does the angular acceleration of the mass remain constant during its motion? If not then how does the angular acceleration change? Is the angular acceleration ever equal to zero when pendulum is in motion? Consider a simple pendulum. Does the angular acceleration of the mass remain constant during its motion? If not then how does the angular acceleration change? Is the angular acceleration ever equal to zero when pendulum is in motion? Consider a simple pendulum. Does the angular acceleration of the mass remain constant during its motion? If not then how does the angular acceleration change? Is the angular acceleration ever equal to zero when pendulum is in motion?Explanation / Answer
1) Simple pendulum:
The potential energy as a function of angle a is:
U(a) = mgL(1 - cos(a))
so the torque is:
-dU/da = -mgL(-)(-sin(a)
= -mgLsin(a)
and the angular momentum is:
(mL^2)da/dt
So:
(mL^2)d2a/dt2 = -mgLsin(a)
d2a/dt2 = - (g/L)*sin(a) ~ - (g/L)*a
which has solution
a(t) = sin(t*sqrt(g/L))
a'(t) = sqrt(g/L)*cos(t*sqrt(g/L))
a''(t) = - (g/L)*sin(t*sqrt(g/L))
1a) Does the angular acceleration remain constant? No, it is sinusoidal with amplitude g/L.
1b) Is the angular acceleration ever = 0 when the pendulum is in motion? Yes, it is 0 whenever sin(t*sqrt(g/L) = 0, at which time a' = sqrt(g/L). That is exactly when a = 0.