After its release at the top of the first rise, a...

After its release at the top of the first rise, a rollercoaster car moves freely with negligible friction. The roller coaster shown in Figure $\mathrm{P} 8.26$ has a circular loop of radius $20.0 \mathrm{~m}$. The car barely makes it around the loop: At the top of the loop, the riders are upside down and feel weightless. (a) Find the speed of the roller coaster car at the top of the loop (position 3$)$. Find the speed of the roller coaster car (b) at position 1 and (c) at position 2. (d) Find the difference in height between positions 1 and 4 if the speed at position 4 is $10.0 \mathrm{~m} / \mathrm{s}$

After its release at the top of the first rise, a rollercoaster car moves freely with negligible friction. The roller coaster shown in Figure $\mathrm{P} 8.26$ has a circular loop of radius $20.0 \mathrm{~m}$. The car barely makes it around the loop: At the top of the loop, the riders are upside down and feel weightless. (a) Find the speed of the roller coaster car at the top of the loop (position 3$)$. Find the speed of the roller coaster car (b) at position 1 and
(c) at position 2. (d) Find the difference in height between positions 1 and 4 if the speed at position 4 is $10.0 \mathrm{~m} / \mathrm{s}$

Key Concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. In the context of a roller coaster, the conversion between kinetic energy and gravitational potential energy determines the speed of the car at various positions, especially when analyzing motion along slopes and loops.

Circular Motion and Centripetal Force

For motion along a circular loop, such as at the top of a roller coaster loop, centripetal force is necessary to keep the car in circular motion. At the top of the loop, the condition for weightlessness occurs when gravity alone provides the centripetal force, which sets a minimum speed necessary for the car to complete the loop safely.

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces like friction, the total mechanical energy (the sum of potential and kinetic energy) remains constant throughout the motion. In roller coaster problems, it allows for the calculation of speeds at various positions on the track by equating energy at different heights.

Gravitational Potential Energy

Gravitational potential energy, which depends on an object's height above a reference level, plays a crucial role in roller coaster dynamics. As the coaster ascends and descends, this potential energy is converted into kinetic energy and vice versa, allowing calculation of speeds at different points along the track.

Transcript

00:02 Okay, so we're giving a little loop -to -loop problem using conservation of energy and centripetal forces.

00:08 We will find out what the speed is right here for the first part.

00:12 And what we know is that f is equal to m .a.

00:16 By newton's second law.

00:18 And in this case, acceleration is going to be gravity, and the force is centripetal.

00:23 So we can replace it with the respective values for force.

00:26 The centripetal one is going to be mv squared over r and mass times acceleration this time it's going to be the acceleration of gravity so this is our basic equation and we want to go ahead and isolate for r or excuse me the velocity so we're going to get that v squared is equal to gr after rearranging and then we just take the square root of that to isolate v so like this notice how mass cancels in this formal just because we have mass on the same side.

01:08 Substituting in the values, we get that the velocity is 14 meters per second at the top.

01:15 To find the second energy, we're going to say that the initial is equal to the final energy.

01:23 And at these points, there will be both kinetic and potential energy.

01:27 So we're going to call this k1 plus u1 is equal to k2 plus plus u2.

01:39 Since we are trying to find the velocity of this one, we'll go ahead and move over the potential energy.

01:46 So we have k1 is equal to k2 plus u2 minus u1.

01:55 So the difference of the potential energies plus the kinetic energy is going to equal the initial kinetic energy.

02:04 Now if we substitute the formulaic versions, we're going to get one half and then v squared.

02:10 We're just going to drop the masses because these are all the same in this equation.

02:16 And then we get the other one -half v squared for the other speed.

02:24 And then label that with a v2.

02:27 And then we have mgh, but again, no m for mass.

02:33 So we're just going to factor out a cheat.

02:36 And we have the second height minus the first height.

02:42 And then we'll go ahead and multiply by 2.

02:44 And square root it.

02:47 So we get that v is equal to the square root of v2 squared plus two chi and the difference of the heights.

03:05 So when we plug in those values, we find that the velocity at the bottom is 31 .3 meters per second.

03:18 So that's part b.

03:21 And now moving on to part c.

03:23 Where we're as to find what is the speed at position two.

03:29 So once again, we're going to equate the two energies...

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