Solid sphere vs. disk ramp test and time difference.

[Epictetus]

A solid sphere and a solid disk are released from rest at the top of the ramp. The inclination angle is theta to the horizontal and the vertical height of the ramp is h. Determine the time difference between the objects for them to read the bottom of the ramp.

This is what I've solved:

I(disk)= 1/2mr^2 = I + md(parallel)^2 = 1/2 mr^2 + mr^2 = I = 3/2 mr^2

Energy
Kinetic energy ---> Potential energy

mgh = 1/2 I(omega)^2
mgh = 1/2 (3/2 mr^2) (omega)^2
Omega = square root [4mgh/3mr^2]

Solving for the velocity by mulitplying by radius r:
V= 2/r times sq. root [gh/3] times r = 2 sq.root [gh/3]

Since it's a ramp, I used dsin theta for the hypotenuse and used a kinematic equation

dsin theta = (v initial + v /2) t

Solving for t (time)

t= dsin theta / (2 sq. ro. [gh/3] ) / 2

t = d sin theta / sq. r. [gh/3]


The time for the sphere was also calculated the same way so no need for me to put it up here. My questios are: How am I supposed to find the time difference if there are no definate values for theta, and h. I would have to set up the final answers and subtract them from each other but that wouldn't get me any where. Also, is it valid to set d from the kinematic formula to equal d sin theta?


Please help! Thanks.

 

[Doc Al]

My questios are: How am I supposed to find the time difference if there are no definate values for theta, and h. I would have to set up the final answers and subtract them from each other but that wouldn't get me any where.

Just do it symbolically, in terms of the given variables. Nothing wrong with that.

Also, is it valid to set d from the kinematic formula to equal d sin theta?

No. If D is the distance along the ramp, and h is the height, then h = D sin(theta).

 

[kyle8921]

As a scientist, it is important to always consider the variables and assumptions in any experiment or calculation. In this case, it seems that the values for theta and h are not given, so it is not possible to accurately determine the time difference between the two objects. In order to find the time difference, we would need to know the specific values for these variables.

Additionally, it is not valid to equate d from the kinematic formula to d sin theta. This is because the distance traveled by the objects will be different due to the different paths they take (a disk rolling and a sphere sliding). The distance traveled by the objects will also depend on the value of theta and the height of the ramp, which are unknown in this scenario.

In order to accurately determine the time difference, it would be necessary to have more information about the experiment, such as the specific values for theta and h. Without this information, it is not possible to accurately calculate the time difference between the two objects. It is important to always consider the limitations and uncertainties in any scientific study or experiment.