A 3.00-cm-long spring has a small plastic bead glued to each end....

Question

A 3.00-cm-long spring has a small plastic bead glued to each end. Charging each bead to −25 nC expands the spring by 0.50 cm. What is the value of the spring constant?

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. A nanoscale tuning fork has two prongs with a initial distance of 4.20 centimeters between them. A small crystal sphere is attached to each prong when a charge of negative 35 nano columns is applied to each sphere, the distance between the prongs increases by 0.68 centimeters, calculate the spring constant of the tuning fork. So that is our end goal is to calculate the spring constant of the tuning fork. OK. So we're given some multiple choice answers here and they're all in the same units of newtons per meter. So let's read them off to see what our final answer might be. A is 0.45 new 0.45 B is 0.68 C is 1.25 and D is 2.50. OK. So let us begin to solve for this problem by writing down all of our known variables. So the initial distance between the prongs. Let's call it I subscript zero is equal to 4.20 centimeters which we need to convert that to meters. So it'd be 4.20 multiplied by 10 to the power of negative m. And the distance increased after charging the spears, let's call that delta X. So this is the distance increased after charging the spears is equal to 0.68 centimeters. So let's convert that to meters really quick. 0.68 multiplied by 10 to the power of negative 2 m. OK. So the charge on each sphere, so we have the charge on the left sphere and then we have the charge on the right sphere is equal to negative 35 nano columns, which is 35 negative 35 multiplied by 10 to the power of negative nine columns when we convert nano columns to columns. So now that we know that the charge on the left sphere and the charge on the right sphere are both negative 35 nano columns or negative 35 multiplied by 10 to the power of negative nine columns. So note we can find the final distance between the prongs. And let's call that R subscript capital L capital R. So RLR by adding the initial distance and the increased distances. OK. So R subscript capital L capital R is equal to I subscript zero plus delta X. So let's plug in all of our known variables to solve for RLR which is the final distance between the prongs. So the final distance between the prongs is equal to 4.20 multiplied by 10 to the power of 2 m plus 0.68 multiplied by 10 to the power of negative 2 m. So the final distance between the prongs is equal two when we plug into a calculator and 0.048. OK. So it's 0.0 or 8 8 m. OK. So now we can determine the spring constant K by recalling and using the equation for Kums Law and the spring constant. So combining these two equations. So when we combine both equations, we determine that the spring constant K is equal to Kums constant multiplied by the charge of the left sphere, the absolute value of the charge of the left sphere multiplied by the absolute value of the right sphere, all divided by the final distance between the prongs squared multiplied by the increased distance. So the distance increased after charging the spheres. OK. So now we can plug in our known variables. And so for K the spring constant, so let's do that. So the numerical value for Kums constant is 9.0 multiplied by 10 to the power of nine. And it's in units of newtons multiplied by meters squared divided by Coolum squared. Or I should say Newton is multiplied by meters squared per Coolum squared multiplied by the absolute value of negative 35 multiplied by 10 to the power of negative nine columns squared VIAS the charge on the left sphere and the charge on the right sphere are equal to each other. All divided by the distance. The final distance between the prongs which was 0.0488 m squared multiplied by delta X which was 0.68 multiplied by 10 to the power of negative 2 m. So when we plug that into a calculator, we should get that. The spring constant is equal to 0. newtons per meter, which is our final answer. So the approximate value for the spring constant of the tuning fork is 0.68 nano sorry 0.68 newtons per meter. Oh, we did it. So that means looking at our final, looking for a final answer out of the multiple choice answers. It has to be the letter B 0.68 newtons per meter. 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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A 3.00-cm-long spring has a small plastic bead glued to each end. Charging each bead to −25 nC expands the spring by 0.50 cm. What is the value of the spring constant?

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