A light, non-conducting straw of length is pivoted at its center so that it can…
PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Torque and Rotational Dynamics · Torque and Rotational Dynamics
Problem
A light, non-conducting straw of length 30cm is pivoted at its center so that it can rotate freely in the horizontal plane with negligible friction. A small neutral metallic sphere of mass 2.0g is glued to one end of the straw. The other end of the straw carries negligible mass.
A charged plastic rod is brought parallel to the straw and positioned so that its tip is 4.0cm from the center of the metallic sphere along the straw’s axis. The rod is held fixed in this position. The metallic sphere polarizes; for the purpose of analysis, treat the induced charges as two point charges of equal magnitude qind separated by 1.0cm along the straw. The nearer induced charge is 4.0cm from the rod, while the farther induced charge is 5.0cm away.
Assume the straw was initially at rest and horizontal before the rod was introduced, and neglect any gravitational torque about the pivot.
(a) Using the induced-charge model, obtain an expression for the net torque τ about the pivot produced by the rod on the metallic sphere. Express τ in terms of qind, the rod’s charge Q, the given distances, and the Coulomb constant k.
(b) Determine the initial angular acceleration of the straw–sphere system immediately after the rod is brought into position. Express your answer symbolically in terms of τ and the moment of inertia I of the system, then compute a numerical value given that the straw’s mass is negligible compared with the 2.0g sphere.
(c) Calculate the numerical value of τ and the corresponding angular acceleration if qind=3.0nC and Q=5.0μC.
(d) Discuss qualitatively how doubling the separation between the two induced charges (while keeping their magnitude the same) would affect both the torque and the angular acceleration. Base your discussion on the form of the expression found in part (a).