A uniform hoop and a disk, having the same mass and the same radius, are released simultaneously from rest at

the top of a ramp. Which object reaches the bottom of the ramp first? The ramp is 1.0meter high. The incline of the plane is 30 degrees. Assume no friction. Show mathematically.

The disc reaches the bottom first, no matter what the ramp height and angle is. Acceleration depends upon the ratio of moment of inertia to M*R^2.

I understand that but I still have to show it mathematically.

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To determine which object reaches the bottom of the ramp first, we need to compare their acceleration down the ramp.

First, let's set up our coordinate system. Let the positive x-axis be directed down the ramp, and the origin be located at the top of the ramp. The height of the ramp is given as 1.0 meter, and the incline of the plane is 30 degrees.

Now, let's analyze the motion of each object separately.

1. Uniform Hoop:
The uniform hoop can be thought of as a series of point masses connected together. For a hoop rolling down an inclined plane without friction, the acceleration down the ramp can be found using the following equation:

a_hoop = g * sin(θ) / (1 + (k^2 / R^2))

Where:
- a_hoop is the acceleration of the hoop down the ramp.
- g is the acceleration due to gravity (approximated as 9.8 m/s^2).
- θ is the angle of the incline (30 degrees).
- R is the radius of the hoop.
- k is the radius of gyration of the hoop, which is equal to R for a hoop.

2. Disk:
Similarly, for a solid disk rolling down an inclined plane without friction, the acceleration down the ramp can be found using the following equation:

a_disk = g * sin(θ) / (1 + (1/2)^2)

Where:
- a_disk is the acceleration of the disk down the ramp.
- θ is the angle of the incline (30 degrees).

Now we can compare the calculated accelerations of the hoop and disk.

Let's assume that the mass and radius of both the hoop and disk are the same, so the effect of mass cancels out.

Substituting the given values into the equations, we find:

a_hoop = (9.8 m/s^2) * sin(30°) / (1 + (1^2 / R^2))

a_disk = (9.8 m/s^2) * sin(30°) / (1 + (1/2)^2)

To determine which object reaches the bottom first, we need to compare their accelerations.

By comparing the equations, we can observe that the acceleration of the hoop is greater than the acceleration of the disk. This means that the hoop reaches the bottom of the ramp first.

Therefore, the uniform hoop reaches the bottom of the ramp first.