What is the maximum tension possible in a 1.00-mm-diameter nyl...

Question

What is the maximum tension possible in a 1.00-mm-diameter nylon tennis racket string? If you want tighter strings, what do you do to prevent breakage: use thinner or thicker strings? Why? What causes strings to break when they are hit by the ball?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A base string made of steel with a diameter of 1.2 millimeters can withstand a maximum force of 570 newtons before breaking wood using a thinner or thicker string helped prevent snapping under high tension. And why does it snap when plucked hint? The ultimate tensile strength of steel is about 5.0 multiplied by 10 to the power of eight newtons per meter squared. Awesome. So our end goal is we're ultimately trying to figure out if using a thinner or thicker string will help prevent snapping under high tension. And then we're also trying to figure out why does it snap when it's plucked? Awesome. So now that we have an idea of what we're solving for, let's read off our multiple choice answers to see what our final answer might be. A is thinner strings would help prevent snapping because they have less mass B is thicker strings would help prevent snapping. Because they are able to withstand greater force due to increased cross sectional area. C is, the thickness of the string does not affect its ability to withstand tension. The frequency of oscillation after the impact causes the snap and finally, D is thinner strings would help prevent snapping because they can vibrate more freely. OK. So first off, let us note that the force a string can withstand before it breaks is directly related to its cross sectional area. We also need to recall and note that the maximum force that the string can withstand before breaking can be written as FM, where FM denotes the max force divided by capital A where capital A denotes the cross sectional area is equal to 5.0 multiplied by 10 to the power of eight and its units are newtons per meter square and this is our tensile strength. So our maximum force. So the max force divided by our cross sectional area is equal to the tensile strength. So now we need to rearrange this equation to isolate and solve for F max. So when we do, we find that F max is equal to drum roll 5.0 multiplied by 10 to the power of eight newtons per meter squared multiplied by A where A once again is our cross sectional area. So let us quickly note, let's make a little side note here that a our cross sectional area is the area of a circle due to the circular shape of the guitar string. So we could say that A is equal to pi multiplied by D divided by two square. Awesome. So let us now plug in two different values for D the diameter to see if a thicker or thinner string dynam Meer could withstand a greater value of the max force. So considering that let us use 1.2 millimeters for the thinner string diameter and 2.4 millimeters for the thicker strings diameter. So for D equals 1.2 millimeters, let us note that F max, so F max FM comma 1.2 millimeters is equal to 5.0 multiplied by 10 to the power of eight newtons per meter square, multiplied by pi multiplied by 1.2 multiplied by 10 to the power of negative 3 m divided by two square is equal to. And when we plug this into our calculator, we will get 565 point or 867. And don't forget that the units are newtons. So similarly, for D equals 2.4 millimeters, we could write that our max force FM comma 2.4 millimeters is equal to 5.0 multiplied by 10 to the power of eight newtons per meter square multiplied by pi multiplied by 2.4. And note that we had to convert millimeters to meters in both cases. So you take 2.4 multiplied by 10 to the power of negative three, they convert millimeters to meters divided by two square is equal to 2261.9467 newtons. Thus, as we can see the maximum force that the steel cable or the steel string can withstand is greater for a larger diameter D values for thicker strings. Let us also recall and note that a string will snap if the tension is greater than F max. Therefore, if a thicker string is used, it would be with able to withstand a greater force due to the increased value of the cross sectional area, which therefore, as a result will prevent the string from breaking or snapping. Thus, the correct answer has to be the letter B I really did it. So once again, we know that our final answer has to be the letter B, thicker strings would help prevent snapping because they are able to withstand greater force due to increased cross sectional area. And that's it. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Uh

What is the maximum tension possible in a 1.00-mm-diameter nylon tennis racket string? If you want tighter strings, what do you do to prevent breakage: use thinner or thicker strings? Why? What causes strings to break when they are hit by the ball?

Key Concepts

Tension in Strings

Material Properties of Nylon

String Thickness and Breakage

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