Energy of a Robotic Arm

MediumPhysElo Quiz #1

14.Energy of a Robotic Arm

+10

A mechanism operating on a horizontal frictionless surface consists of two uniform rods, each of mass

m

and length

L

. The first rod is attached to a fixed pivot. The second rod is attached to the free end of the first rod by a hinge. A torsion spring of spring constant

\kappa

at the hinge is relaxed when the two rods are collinear and pointing in the same direction. At a certain instant, the first rod makes an angle

\theta_1

with a fixed reference axis and is rotating with angular velocity

\omega_1

. The second rod makes an angle

\theta_2

with the same axis and is rotating with angular velocity

\omega_2

. Which equation correctly describes the total energy of the system?

E = mL^2 \left( \frac{1}{6}\omega_1^2 + \frac{1}{6}\omega_2^2 \right) + \frac{1}{2}\kappa(\theta_1-\theta_2)^2

E = mL^2 \left( \frac{2}{3}\omega_1^2 + \frac{1}{6}\omega_2^2 \right) + \frac{1}{2}\kappa(\theta_1-\theta_2)^2

E = mL^2 \left( \frac{2}{3}\omega_1^2 + \frac{1}{6}\omega_2^2 + \frac{1}{2}\omega_1\omega_2\cos(\theta_1-\theta_2) \right) + \frac{1}{2}\kappa(\theta_1+\theta_2)^2

E = mL^2 \left( \frac{2}{3}\omega_1^2 + \frac{1}{6}\omega_2^2 + \frac{1}{2}\omega_1\omega_2\cos(\theta_1-\theta_2) \right) + \frac{1}{2}\kappa(\theta_1-\theta_2)^2

E = mL^2 \left( \frac{2}{3}\omega_1^2 + \frac{1}{6}\omega_2^2 + \omega_1\omega_2\cos(\theta_1-\theta_2) \right) + \frac{1}{2}\kappa(\theta_1-\theta_2)^2

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