A pendulum is made by tying a 500 g ball to a 75-cm-long string. ...

Question

A pendulum is made by tying a 500 g ball to a 75-cm-long string. The pendulum is pulled 30 degree to one side, then released. What is the ball's speed at the lowest point of its trajectory?

Accepted Answer

Everyone in this problem. A roller coaster car is released from a height of 50 m and reaches the bottom of the truck. We're asked to find the velocity of the car at the bottom of the track. If it started with an initial velocity of 0 m per second, we're given four answer choices all in meters per second. Option A 89 option B option C 31.3 and option D 33.1. So we have a roller coaster released from some height and we kind of have two things going on here. We have this change in height when the roller coaster goes from the top down to the bottom and we also have this velocity. So what we can think about here is the conservation of mechanical energy, ok? Now, conservation of mechanical energy tells us that the initial kinetic energy, the initial potential energy, it's gonna be equal to the final kinetic energy plus the final potential energy. Ok? We have no net external forces acting on this roller coaster. We're gonna assume that it's nice and simple and so we can consider this conservation of mechanical energy. Now, when we talk about this um potential energy, both initially and finally, we don't have any springs here. We only have the energy due to gravity. OK. So we have the gravitational potential energy and so these potential energies we can rewrite them. OK? Just as UG for each. OK. So we have K knot plus UG knot is equal to KF plus UGF. OK. Where the subscript knot is gonna indicate the initial time point when the roller coaster is at the top. And then the F subscript is gonna indicate that final time point when the roller coaster is at the bottom of the truck. All right. Now, kinetic energy recall is given by one half MV squared and the potential energy gravitational is gonna be MGH. So we can rewrite our equation using that information and it's gonna be one half M V not squared plus MGH knot is equal to one half MVX word plus MGHF. All right. And what we're interested in finding here is the velocity of the car at the bottom. So that VF value now, we're told that the car is released or starts with an initial velocity of m per second. That means that V not at zero. And so the entire first term is gonna go to zero. Now, at the end, the roller coaster reaches the bottom of the track. If we consider the bottom of the track to be height zero, then the final height is going to be zero. And so the last term on the right hand side is going to be zero. So what we're left with is MGH knot is equal to one half MVF squared. We can divide both sides by M and then we have GH not is equal to one half VF squared. Let's go ahead and substitute our in our numbers. So the acceleration due to gravity G is gonna be 9.8 m per second squared. The initial height, ok. We're told that the roller coaster is released from a height of 50 m. So the initial height is going to be 50 m and this is gonna equal one half multiplied by VF squared. That velocity we're looking for now VF squared, we can write, OK, we can simplify. We're gonna have 9.8 multiplied by 50. We're gonna multiply by two to get rid of this one half and that's gonna give us 980 ok? And we have meters squared per second squared. And when we take the square root, we get that, that velocity is gonna be about 0.305 m per second. All right. So rounding to one decimal place, we found that the speed of the car at the bottom of the track is gonna be 31.3 m per second, which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

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A pendulum is made by tying a 500 g ball to a 75-cm-long string. The pendulum is pulled 30 degree to one side, then released. What is the ball's speed at the lowest point of its trajectory?

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