A 56-kg student runs at 6.0 m/s, grabs a hanging 10.0‑m-long r...

Question

A 56-kg student runs at 6.0 m/s, grabs a hanging 10.0‑m-long rope, and swings out over a lake (Fig. 8–49). He releases the rope when his velocity is zero. What is the angle θ when he releases the rope?
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Accepted Answer

Welcome back. Everyone in this problem. A monkey runs toward a 12 m long vine that hangs from a tree at a speed of 4 m per second. Upon reaching the vine, it grabs hold and swings on it. It then releases the vine as soon as the vine comes to a stop, given that the mass of the monkey's 46 kg, evaluate at what angle phi with the vertical. It releases the vine for our answer choices. A says 11 degrees B 21 degrees C 30 degrees and D 86 degrees. Now, before we continue, let's try to make note of all the information we have in our problem here. So we're talking about a monkey that's running to a vine that's 12 m long. So let's let L be equal to 12 m. Next, we know that it runs towards the vine at a speed of 4 m per second. So we can say that that's the monkey's initial speed. So V I equals 4 m per second. Let's put that on our diagram here. OK? We also know OK that upon reaching the vine, it grabs, hold and swings on it and then it releases the vine as soon as the vine comes to a stop. OK. So since the vine comes to a stop, that means at this point, the final speed, the monkey is gonna be 0 m per second. So again, we can represent on our diagram here. OK? And let me, let me put it below it just to differentiate that I'm talking about speed here. And we also know that the monkey is 46 kg. So let's let m be equal to 46 kg. I know we want to use this information to help us figure out the angle phi with the vertical at which it releases the vine. So in other words, we want to figure out what Phi is equal to. OK. Now let's go to our diagram. What information does our diagram tell us? Well, notice here that our vine or 12 m vine is here, it swings until it comes to a stop. So when it comes to a stop, it will also have the same length of 12 m. So this length L is, is 12 m. OK? And if that's the case, then we can think about a horizontal line. OK? That meets our vertical at the point where the vine stops, that is from the vine that stops. OK. And no, that tells us then that it's going to have a length that is going to be between the top of the vine to where the vine stops. The horizontal form where the vine stops. Now, what's that length going to be? Well, at this point, the monkey is going to be a height of 8 m off the ground. OK. And since it took the vine up from the ground, then that length, that unknown length that we're trying to figure out is going to be the difference between the length of the vine. L let me stay consistent with our symbol here and the height that the monkey is after ground. In other words, that length is going to be equal to L minus H. Now, what do you notice here? Well, at this point, notice that we have a right triangle that's formed with our vine, the vertical and the horizontal where vertical length is L minus H and our hypotenuse of our right triangle is L. So if we want to find Phi, we can use the trigonometric ratios, which trigonometric ratio relates the adjacent side to the hypotenuse will recall that the core sign of an angle came is equal to the adjacent side divided by the hyperten. So what that tells us then? OK, is that the cosine of Phi, let me get rid of that here. The cosine of Phi is going to be equal to our adjacent side, L minus H divided by our hyper tenuous L. We can even expand that a bit further to say that a cosine of Phi is going to be equal to L divided by L which is one minus H divided by L. But before we try to solve for Phi any further, we don't know what is. We don't know how high the monkey is off the ground. So if we can figure out the value of, then we should be able to substitute and eventually solve for Phi. So let's focus on that for a bit. What can we use that can help us? Well, we can use the principle of conservation of energy. Let's think about what happens initially and what happens whenever the monkey lets go of the vine. OK. Now, initially here, when we think about our mechanical energy involved, we'll be thinking about our potential and kinetic energy. But notice that at this point, the initial height of the monkey is zero. So that means that the potential energy is going to be zero. OK. So if we apply the principle of conservation of energy, it implies that the initial energy equals the final energy. But we're saying that initially our monkey would have only kinetic energy. OK. So what that means then is that the initial kinetic energy is going to be equal to the final energy. And at the final energy, like we said, our V comes to a stop, our final velocity is zero, which means it will have no kinetic energy, only potential energy because it's going to be a height of the ground. So our initial kinetic energy is going to be equal to our final potential energy. Applying what we know about kinetic and potential energy. Recall that kinetic energy is equal to half MV squared. And potential energy due to gravity equals the mass multiplied by G, the acceleration due to gravity multiplied by the height which OK, which then tells us by our formula that half MV I squared initial kinetic energy equals MGH. OK. Where H represents the height of the monkey off the ground? That's its potential energy. When the vine stops before it jumps notice we can cancel M from both sides. OK? To tell us then that G equals half V squared. OK. So thus the height is going to be equal to V squared divided by two G. Now that we have an expression for H before we solve, we're not gonna solve yet. But let's substitute it into our formula for the cosine of Phi. OK. To know, OK. Therefore, remember we said the cosine of Phi equals one minus H divided by L. So now that's going to be one minus HV squared divided by two G all divided by L. So we can say that it's also being divided by L here now that we know the length of our VL or initial velocity 4 m per second. And we can take the acceleration due to gravity G to be equal to a 9.81 m per second squared. That means Phi is going to be equal to the inverse cosine of that known expression. And thus to solve, we can substitute our values. So that means Phi then is going to be equal to the inverse cosine of one minus 4 m per second squared divided by two multiplied by G 9.81 m per second squared multiplied by the length of the vine which is 12 m. OK. So now we can put this expression directly into our calculator. When you do that, you should get Phi to be equal to 21.24 degrees. And if we round it to the nearest degrees, like we did the rest of our answers, that means Phi is equal to approximately 21 degrees. In other words, the angle that the vertical makes with the vine when it comes to a stop is 21 degrees. Therefore, B is the correct answer. Thanks for watching everyone. I hope this video helped.

A 56-kg student runs at 6.0 m/s, grabs a hanging 10.0‑m-long rope, and swings out over a lake (Fig. 8–49). He releases the rope when his velocity is zero. What is the angle θ when he releases the rope?
<IMAGE>

Key Concepts

Conservation of Energy

Kinematics of Circular Motion

Trigonometric Relationships

Watch next

A 56-kg student runs at 6.0 m/s, grabs a hanging 10.0‑m-long rope, and swings out over a lake (Fig. 8–49). He releases the rope when his velocity is zero. What is the angle θ when he releases the rope?<IMAGE>

Key Concepts

Conservation of Energy

Kinematics of Circular Motion

Trigonometric Relationships

Watch next

A 56-kg student runs at 6.0 m/s, grabs a hanging 10.0‑m-long rope, and swings out over a lake (Fig. 8–49). He releases the rope when his velocity is zero. What is the angle θ when he releases the rope?
<IMAGE>