The track near the top of your favorite roller coaster has a circular shape with a diameter of 36 m. When you are at the top, you feel as if you weigh only one-fifth of your weight on the ground. What is the speed of the roller coaster?
Why did the roller coaster go on a diet? Because it wanted to weigh less on the track!
But let's get down to the math. We know that you feel one-fifth of your weight on the ground, so we can say that the normal force on you at the top of the loop is four-fifths of your weight. This force is directed towards the center of the circular track, providing the necessary centripetal force for the circular motion.
The equation for centripetal force is Fc = m * v^2 / r, where Fc is the centripetal force, m is mass, v is velocity, and r is the radius/diameter of the loop.
Now, since we know that the gravitational force is the weight (Fg) and is equal to m * g, we can say that Fg = m * g and Fc = 4/5 * Fg.
Substituting these into the centripetal force equation, we have 4/5 * Fg = m * v^2 / r.
With a bit of rearranging, we get v^2 = (4/5 * Fg * r) / m.
Now, we need to find v. We know that Fg is the weight, and your weight on the ground is m * g. So, your weight on the top of the loop is one-fifth of your weight on the ground, which is (1/5) * m * g.
Substituting this into the equation, we have v^2 = (4/5 * (1/5) * m * g * r) / m.
Canceling out the m's and simplifying, we get v^2 = 4/25 * g * r.
Finally, we take the square root of both sides to find the speed of the roller coaster: v = √((4/25 * g * r)).
So, plug in the values: g = 9.8 m/s^2 (acceleration due to gravity), and r = 36 m (diameter of the loop).
Calculating, we get v ≈ 7 m/s (rounded to one decimal place).
Therefore, the approximate speed of the roller coaster is 7 meters per second. Enjoy the ride, but don't forget to keep your arms and legs inside the cart at all times!
To find the speed of the roller coaster, we can use the centripetal force formula:
F = (m * v^2) / r
Where:
F is the force experienced by the person (weight)
m is the mass of the person
v is the velocity
r is the radius of the circular track
We know that at the top of the roller coaster, the person feels as if their weight is one fifth of their weight on the ground. Let's call the weight of the person on the ground W.
So, the weight of the person at the top of the roller coaster is 1/5 * W.
Now, let's substitute the formula for weight:
F = m * g,
where g is the acceleration due to gravity (which is approximately 9.8 m/s^2).
At the top of the roller coaster, the net force acting on the person is the centripetal force F:
F = m * g = (m * v^2) / r
Rearranging the formula, we get:
v^2 = (F * r) / m
Substituting F = 1/5 * W and isolating v, we have:
v = √((F * r) / m)
We can plug in the known values:
r = 36 m (diameter of the track, so radius is half that)
F = 1/5 * W (one-fifth of the weight on the ground)
m = the mass of the person (which is not given)
Unfortunately, we don't have the value of m (mass of the person), so we cannot calculate the speed of the roller coaster without that information.
To find the speed of the roller coaster, we need to use the concept of centripetal force. At the top of the roller coaster loop, the centripetal force required to keep the riders moving in a circular path is provided by the normal force (the force exerted by the track on the rider) and the gravitational force (the weight of the rider).
Let's break down the problem step by step:
1. Convert the weight on the ground to the mass of the rider:
Since weight is a force and is given by W = mg, where m is the mass of the rider and g is the gravitational acceleration (approximately 9.8 m/s^2), we can set up the equation:
W_ground = mg
From the given information, the weight on the top of the roller coaster is only one-fifth of the weight on the ground. So:
W_top = (1/5) * W_ground = (1/5) * mg
2. Determine the net force at the top of the roller coaster:
The net force at the top of the roller coaster is the difference between the gravitational force (downwards) and the normal force (upwards):
Net force = Gravitational force - Normal force
F_net = mg - N
Here, the normal force N is the force exerted by the track on the rider at the top of the roller coaster.
3. Relate the net force to the centripetal force:
The net force at the top of the roller coaster is equal to the centripetal force required to keep the rider moving in a circular path. The centripetal force is given by F_c = (mv^2) / r, where m is the mass of the rider, v is the speed of the roller coaster, and r is the radius of the circular path.
4. Equate the net force and the centripetal force to find the speed v:
Equate the net force (mg - N) to the centripetal force (mv^2 / r):
mg - N = (mv^2) / r
Substituting the weight on the top (W_top = (1/5) * mg) and the radius of the circular path (r = diameter/2 = 36/2 = 18 m), we have:
mg - N = (mv^2) / 18
Now, we need information about the normal force at the top to proceed further. Unfortunately, the given problem doesn't provide any direct information about the normal force. More information is needed to solve the problem.
apparent weight= 1/5 mg
but apparent weight= weigh-centripetalforce
1/5 mg= mg-mv^2/r
solve for v