Rotational Motion: Speed of 13kg Cylinder on 30° & 60° Inclines

[Soaring Crane]

A 13 kg cylinder with a .54 m diameter rolls without slipping down a 30 degree incline from a height of 1.25 m.

a. If the cylinder has I = (mr^2)/2, what will its speed be at the incline's base?

After using the conservation of mechanical energy I got v = sqrt[(4/3)gh], which is 4.04 m/s.

b. What is its speed if it rolls from the same height down a 60 degree incline?

Now wouldn't the speed be the same as in part a since v is not dependent on the angle?

My major question is why are both speeds the same if the angle changes?

If my calculations are incorrect, please inform me. Thank you for any help.

 

[Doc Al]

After using the conservation of mechanical energy I got v = sqrt[(4/3)gh], which is 4.04 m/s.

Right!

b. What is its speed if it rolls from the same height down a 60 degree incline?

Now wouldn't the speed be the same as in part a since v is not dependent on the angle?

Right.

My major question is why are both speeds the same if the angle changes?

Since the initial PE is the same in both cases, the final speed will be the same. What does change is the acceleration down the incline. The smaller angle produces a smaller acceleration, but the distance is greater by the same factor, so the final speed remains the same.

 

[wais]

Your calculations are correct. The speed of the cylinder at the base of the incline will be the same in both cases, regardless of the angle. This is because the speed of a rolling object is not affected by the angle of incline. The speed only depends on the height from which the object is released and the rotational inertia (I) of the object.

In this case, the rotational inertia (I) of the cylinder is (13kg)(0.27m)^2/2 = 0.729 kgm^2. Plugging this into the equation v = sqrt[(4/3)gh], we get v = sqrt[(4/3)(9.8m/s^2)(1.25m)] = 4.04 m/s.

To understand why the speed is the same in both cases, we can look at the forces acting on the cylinder. When the cylinder is released, it experiences a downward force due to gravity and a normal force from the incline. These two forces create a torque that causes the cylinder to accelerate and roll down the incline. The angle of the incline does not affect the magnitude of these forces, only their components in the direction of motion.

Since the speed of a rolling object is determined by its rotational inertia and the forces acting on it, and the forces are the same in both cases, the speed will also be the same.

I hope this helps clarify your understanding of rotational motion and the relationship between speed and angle of incline. Keep up the good work with your calculations!