You have a lightweight spring whose unstretched length is 4.0 cm....

Question

You have a lightweight spring whose unstretched length is 4.0 cm. First, you attach one end of the spring to the ceiling and hang a 1.0 g mass from it. This stretches the spring to a length of 5.0 cm. You then attach two small plastic beads to the opposite ends of the spring, lay the spring on a frictionless table, and give each plastic bead the same charge. This stretches the spring to a length of 4.5 cm. What is the magnitude of the charge (in nC) on each bead?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem, two small plastic beads with equal charges are attached to a 10.0 centimeter long uns stretched rubber band causing it to stretch to a length of 13.0 centimeters due to electrostatic repulsion. The rubber band is stretched to 15.0 centimeters in a separate experiment by applying a force equivalent to a 0.1 g mass hanging vertically, determine the magnitude of the charge and nano columns on each bead. OK. So our end goal is to calculate the magnitude of the charge and nano columns on each beat. OK. So we're given some multiple choice answers here. They're all in the same units of nano columns. Let's read them off to see what our final answer might be. A is 14.33 B is 33.23 C is 12.66 and D is 16.58. OK. So first off, let us solve for the rubber band spring constant K. So, so the spring constant will equal K and the weight of the 0.1 g mass. So 0.1 g mass is the same as 0. kilogram stretches. The rubber band from 15 centimeters from an uns stretched length of 10 centimeters. So that means there's a five centimeter stretch. Ok. So that is the same as saying 0.050 m stretch. OK. So now we must recall that the force is equal to the mass multiplied by the gravity. So let's plug in our known values to solve for the force. So the force is equal to the mass which we determine to be 0. kg multiplied by gravity which is 9. newtons per kilogram. The kilograms cancel out leaving newtons. And when we plug into a calculator, we should get 0. newtons. Now we need to recall and use Hooks Law to solve for the spring constant K. So the spring constant K to help us solve for that, we need to find, we need to say that the force is equal to the spring constant multiplied by delta X. So now we need to rearrange this equation to solve for the spring constant on its own. So K is equal to the force divided by delta X where delta X is the stretch of the rubber band. So when we plug in our known values, we determine the force to be 0. newtons divided by the stretch in the rubber band which we determine to be 0. m. So when we plug that into a calculator, we should get that. The spring constant K is equal to 0. newtons per meter. Awesome. So now we need to balance the repulsive Coolum force and the restoring spring force on either bead, the rubber band is stretched from its uns stretched length of centimeters to 13 centimeters, which is a stretch of three centimeters. Now, we can use Kums Law and set it equal to Hooks Law. So let's do that. So s constant multiplied by the charged by the charge squared divided by the distance between the beats squared is equal to the spring constant multiplied by the stretch of the rubber band. So now we need, can, now we need to take this equation and rearrange it using some algebra to isolate and solve for Q the charge. So Q, the charge is equal to the square root of the spring constant multiplied by the stretch of the rubber band multiplied by the distance between the beats squared divided by is constant. So now we can plug in all of our known variables to solve or Q the charge. So let's do that. So the charge is equal to the square root of the spring constant, which is 0. newtons per meter multiplied by the stretch of the rubber band, which in this case is 0.030 m means it's three centimeters converted to meters multiplied by the distance between the beads squared, which is 0. m because it's 13 centimeters converted to meters squared, all divided by Kums constant, which is 9.0 multiplied by 10 to the power of nine newtons multiplied by meter squared per Coolum squared. So Q equals the square root of all that. So when we take the square root of all of that and put it into a calculator, we should get that Q is equal to 3. two, 28, multiplied by 10 to the power of negative eight columns. But we need to multiply it by 10 to the power of nine to convert it from columns to nano columns. So when we do that, we should get 33. three nano columns, which is our final answer. Ray, we did it. So going back to look at our multiple choice answers. That means the correct answer has to be the letter B 33. Nano Polis. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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