A ball with a net charge of +450 nC and mass 0.75 g is suspended from a thread of length 12 cm i

A ball with a net charge of +450 nC and mass 0.75 g is...

A ball with a net charge of $+450 \mathrm{nC}$ and mass 0.75 g is suspended from a thread of length $12 \mathrm{cm}$ in a uniform electric field of $8.0 \mathrm{kV} / \mathrm{m}$ downward. Do not ignore gravity. The system acts like a pendulum but with a downward electric force added to the gravitational force. The ball is released from rest when the thread makes an angle of $32^{\circ}$ with the vertical. How fast is it moving when the thread is vertical?

A ball with a net charge of $+450 \mathrm{nC}$ and mass 0.75 g is suspended from a thread of length $12 \mathrm{cm}$ in a uniform electric field of $8.0 \mathrm{kV} / \mathrm{m}$ downward. Do not ignore gravity. The system acts like a pendulum but with a downward electric force added to the gravitational force.
The ball is released from rest when the thread makes an angle of $32^{\circ}$ with the vertical. How fast is it moving when the thread is vertical?

Key Concepts

Electric Potential Energy

Electric potential energy arises when a charged object is placed in an electric field. The work done by or against the electric force during displacement results in changes in the electric potential energy. In this problem, the uniform electric field adds an additional force component, so the electric potential energy change, along with the gravitational potential energy change, is taken into account in the energy balance.

Pendulum Dynamics

Pendulum dynamics involves the motion of an object suspended from a pivot, which swings under the force of gravity. When additional forces, such as those due to a uniform electric field, are present, the analysis still relies on energy considerations, but must include the contributions of all forces along the motion's path to accurately determine the speed or position of the pendulum bob.

Energy Conservation

This concept refers to the principle that the total mechanical energy of a system (the sum of kinetic and potential energies) remains constant if only conservative forces are doing work. In the context of this problem, the change in both gravitational and electric potential energy is transformed into kinetic energy of the moving ball.

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. For a pendulum, changes in vertical height directly affect its gravitational potential energy, and when the pendulum swings, the loss or gain in this energy contributes to the kinetic energy of the moving object.

Transcript

00:01 Here we're going to see what happens to a pendulum if it has a bob with a mass m as well as a charge, and that charge is in an electric field that points down along with gravity.

00:16 The question is, after the pendulum is released from rest, what speed does it have when it reaches the bottom? so that is what we're after is the speed.

00:33 And as we would ordinarily solve this problem using conservation of energy, we are going to use that principle, but the energy is going to be both, the potential energy is going to be both gravitational and electric.

00:56 And that electric potential energy is q times the electrical, field times delta, in this case, delta y.

01:14 Probably easier to look at that as q times v to begin with, or delta v.

01:26 It's potential, not velocity.

01:27 I've got to make my velocity look a little bit different than my potential.

01:33 But potential is related to electric field through a spatial relationship.

01:45 Okay, so it's released from rest, the initial position is at height h above the bottom.

01:53 So our initial kinetic energy is going to be zero.

02:00 Our initial potential energy is going to be gravitational, m -g -h, plus electrical, q -e -h.

02:12 So yes, the electricity is just adding a new force in the amount of q times e.

02:20 And that has got to equal to the final kinetic energy plus zero plus zero, because h is going to be equal to zero down at the bottom.

02:36 And that's where we're going to call our reference position for potential energy...

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