a(t) = íAȦ^2 cos (Ȧt + ij)
a(t) = íȦ^2 x(t)
amax = AȦ^2
a = acceleration, A = amplitude
Ȧ = angular velocity
x = displacement, t = time
ij = phase constant
What is the acceleration at t= 1.6
seconds?
a(t) = íAȦ^2 cos (Ȧt + ij)
a(1.6 s) = í7.6 m/s^2
14.9 - A torsional pendulum
The torsional pendulum shown in Concept 1 is another device that exhibits simple
harmonic motion. A torsional pendulum consists of a mass suspended at the end of a
stiff rod, wire or spring. It does not swing back and forth. Instead, the mass at the
bottom is initially rotated by an external torque away from its equilibrium position. The
elasticity of the rod supplies a restoring torque, causing the mass to rotate back to the
equilibrium position and beyond. The mass rotates in an angular version of simple
harmonic motion.
Earlier, we stated that for SHM to occur, the force must be proportional to displacement.
Since torsional pendulums rotate, we must use angular concepts to analyze them. With
a torsional pendulum, a restoring torque, not a force, acts to return the system to its
equilibrium position. The restoring torque is proportional to angular displacement, just
as a restoring force is proportional to (linear) displacement. The moment of inertia of the
system takes on the role that mass plays in linear SHM.
The same analysis that applies to linear displacement, velocity and acceleration applies
equally well to angular displacement, angular velocity and angular acceleration. In
Equation 1, you see an equation that states the nature of the restoring torque. It equals
the negative of the product of the torsion constant and the angular displacement.
Torsional pendulum
Exhibits simple harmonic motion
Use rotational concepts to analyze