Rolling Without slipping and inertias

[ruffrunnr]

I need help with this...

A hollow, thin walled cylinder and a solid sphere start from rest and roll without slipping down an inclined plane of length 3 m. The cylinder arrives at the bottom of the plane 2.4 seconds after the sphere. Determine th angle between the inclined plane and horizontal.

 

[lpfr]

Maybe there is a shorter way to do it. But for now I just tried this method:
Find the angular acceleration taking, as center of rotation the point of contact with the inclined plane. For this you must compute the torque due to the weight and the moment of inertia around the contact point.
You will need to compute this moment of inertia using the well known formula.
Once you have the angular acceleration you can compute the time as a function of the angle. Calculate the total angle to roll down the plan.
Do this for the two objects. The ratio of the two formulas gives you the ratio of two times.
Now you have the ration and the difference between the two times you can compute each one, and then derive the angle of the plan.

 

[m2003]

Rolling without slipping is a phenomenon that occurs when an object simultaneously rotates and translates without any slipping or sliding motion. This type of motion is dependent on the object's inertia and the surface it is rolling on. In this scenario, we have a hollow, thin-walled cylinder and a solid sphere rolling down an inclined plane.

To determine the angle between the inclined plane and horizontal, we can use the given data of the objects' arrival time at the bottom of the plane. Since the cylinder arrives 0.6 seconds earlier than the sphere, we can assume that it has a larger inertia and therefore, a smaller angle of inclination.

Inertia is the resistance of an object to change its state of motion. In this case, the cylinder has a larger inertia due to its hollow, thin-walled structure compared to the solid sphere. This means that it requires more force to accelerate the cylinder and therefore, it takes longer to reach the bottom of the plane.

To calculate the angle, we can use the formula for the acceleration of a rolling object without slipping, which is equal to the acceleration due to gravity multiplied by the sine of the angle of inclination. Using the given data and this formula, we can solve for the angle and determine that it is approximately 17.5 degrees.

In conclusion, the phenomenon of rolling without slipping is dependent on an object's inertia and the surface it is rolling on. In this scenario, the difference in arrival time of the cylinder and sphere can be attributed to their different inertias, which in turn affects the angle of inclination of the inclined plane. This showcases the importance of understanding inertia in studying the motion of objects.