Hollow and solid cylinder, going down a slope

In summary, the problem presents two identical cylinders on identical slopes, one solid and one hollow, being pulled by gravity. The time taken for each cylinder to reach the bottom of the slopes is being deduced, taking into account the rotational inertia and torque of each cylinder. The solution is found by assuming a friction force and calculating the angular and linear acceleration of each cylinder. The possibility of solving the problem without assuming or calculating friction is also mentioned.
  • #1
wavingerwin
98
0

Homework Statement


Well, I thought of the question myself:
There are two identical straight slopes, 300 to the horizontal and 10m long at their hypotenuse. Each of them has one cylinder of identical shape of r=1m and mass (1kg) respectively. However, one is solid whereas the other is hollow.
They are attracted downwards by the same gravity force (10N, suppose gravitational accerleration is 10ms-2), and as a result of the support
force perpendicular to the slope's surface, an identical pull force parallel to the surface is pulling each cylinder (5N, derived from: sin300=F/Fg).
Deduce the time taken for each cylinder to reach the bottom of the slopes!
(Of course, the hollow will take a longer time because it has a greater rotational inertia due to its mass concentrated on further distance compared to the solid cylinder, hence angular acceleration is bigger)

Homework Equations


Rotational inertia of hollow cylinder = mr2
Rotational inertia of solid cylinder = mr2/2
Torque = Rotational inertia*angular acceleration
Torque = force*perpendicular distance
linear acceleration = angular acceleration*radius
linear distance = initial linear speed*time taken + linear acceleration*time taken2/2
(initial linear speed = 0)

3. The attempt at the solution
When i tried to solve this question, i cannot do it without calculating or assuming friction force between the cylinders and the slope's surface. (Hence, the force of 5N which pulls the cylinders is not needed).

Suppose, the friction is 1N, hence torque is 1Nm.
Angular acceleration of the solid cylinder is Torque/rotational inertia = 1/0.5 = 2radians/s
Angular acceleration of the hollow cylinder is Torque/rotational inertia = 1/1 = 1radians/s

Linear acceleration of the solid cylinder = 2ms-2
Linear acceleration of the hollow cylinder = 1ms-2

hence using the last equation and rearrangements, time taken for solid cylinder is 101/2s
and time taken for hollow cylinder is 201/2s

Am i right?
can the problem be solved without assuming/calculating friction?

Thank You!
 
Last edited:
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  • #2
If you don't have friction, you don't have any rotation... only the friction force applies a torque to your cylinder!

If you specify that the friction force is sufficient so that the motion is purely rotational (no slippage), I think you can solve the problem using conservation of energy.
 

1. What is the difference between a hollow and solid cylinder?

A hollow cylinder is a three-dimensional shape with a circular cross-section and empty space in the middle, while a solid cylinder is completely filled with material throughout its entire volume.

2. How does the shape of the cylinder affect its movement down a slope?

The shape of the cylinder plays a crucial role in its movement down a slope. A solid cylinder has a larger mass compared to a hollow cylinder with the same dimensions, leading to a higher inertia and slower acceleration. However, the hollow cylinder may experience more rolling resistance due to its empty space, resulting in a slower movement overall.

3. What factors affect the speed of a cylinder going down a slope?

The speed of a cylinder going down a slope is influenced by several factors, including the angle of the slope, the mass and shape of the cylinder, the surface conditions, and the presence of external forces such as friction or air resistance.

4. How does friction affect the movement of a cylinder down a slope?

Friction is a force that opposes the motion of an object. In the case of a cylinder going down a slope, friction between the cylinder and the surface can slow down its movement and even prevent it from reaching the bottom. The amount of friction depends on the surface conditions and the weight of the cylinder.

5. Can the direction of the slope affect the movement of a cylinder?

Yes, the direction of the slope can significantly impact the movement of a cylinder. For example, a hollow cylinder may roll faster down a slope in the same direction as its open end, while it may face more resistance and roll slower if the slope is in the opposite direction. Additionally, the angle of the slope can also affect the speed and acceleration of the cylinder.

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