Circular motion - velocity of carts on a rollercoaster

[klaw25]

Homework Statement

Six roller-coaster carts pass over the same semicircular "bump." The mass M of each cart (including passenger) and the normal force n of the track on the cart at the top of each bump are given in the figures. Rank the speeds (from largest to smallest) of the different carts as each passes over the top of the bump.
Cart #1: Normal force of 200 N, mass of 400 kg
Cart #2: Normal force of 800 N, mass of 800 kg
Cart #3: Normal force of 400 N, mass of 200 kg
Cart #4: Normal force of 400 N, mass of 100 kg
Cart #5: Normal force of 800 N, mass of 100 kg
Cart #6: Normal force of 300 N, mass of 300 kg

Homework Equations

F = ma
a = v^2 / R

The Attempt at a Solution

If you are supposed to find the acceleration by using F = ma, then these are the accelerations:
#1: 0.5 m/s^2
#2: 1 m/s^2
#3: 2 m/s^2
#4: 4 m/s^2
#5: 8 m/s^2
#6: 1 m/s^2
..I don't know if that's the correct initial step. I'm not sure what to do from here. The radius is not given.

 

[Rake-MC]

Radius is not given, however they do not ask for a numerical answer, just an order of fastest speeds to slowest speeds.

At the peak of the semicircle, you are given the normal force, you are also given the mass of each cart so you can calculate the force due to gravity
When you sum up these two forces, you will have the net force (hopefully it's down or else the carts will fly off the tracks!).

You know that:

$$F = \frac{mv^2}{r}$$

So you can re-arrange for velocity in terms of radius. Repeat for each cart, then simply rank them in order.

 

[baba89]

As a scientist, the first step in approaching this problem is to identify the relevant equations and variables. In this case, we are dealing with circular motion, so the equation a = v^2 / R is relevant. Additionally, we are given the masses and normal forces of the carts, which we can use to calculate the acceleration.

Using the given information, we can calculate the acceleration for each cart using the equation F = ma. For example, for cart #1, we have F = 200 N and m = 400 kg, so a = F/m = 200 N / 400 kg = 0.5 m/s^2.

Next, we need to consider the relationship between acceleration and velocity in circular motion. As the carts pass over the top of the bump, they all have the same radius of curvature. This means that the acceleration is directly proportional to the square of the velocity, as shown in the equation a = v^2 / R. This means that the cart with the largest acceleration will also have the largest velocity.

Using this information, we can rank the carts from largest to smallest velocity based on their calculated accelerations. The ranking would be as follows:

1. Cart #5 (largest acceleration, largest velocity)
2. Cart #4
3. Cart #3
4. Cart #6
5. Cart #2
6. Cart #1 (smallest acceleration, smallest velocity)

In summary, the carts with the largest masses and normal forces (#2 and #5) will have the largest velocities, while the cart with the smallest mass and normal force (#1) will have the smallest velocity. This is because the larger mass and normal force result in a larger acceleration, which in turn leads to a larger velocity.