How Does Cylinder Dynamics Change When Rolling Down a Steep Inclined Plane?

In summary, we can calculate the minimum slope (angle θ) needed for a cylinder to start rolling down a ramp, assuming the forces overcome static friction. We can also calculate the angular and scalar velocity of the cylinder at this point, using the mass, moment of inertia, and slope angle. If the cylinder starts to slip, the acceleration will increase slightly. Other variables such as height (h) and gravity (g) can also be incorporated into the equations.
  • #1
jaumzaum
434
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If we had a cylinder rolling down a ramp, with scalar friction coefficient us and kinetic friction coefficient uc, and we assume for example that the inclination of the ramp is enough to make the cylinder reach the acceleration needed to exceed the scalar friction of rolling. It means that the scalar velocity of the cylinder is not equal to ωR anymore (where ω is the angular velocity). So how can we calculate the angular and scalar velocity now? Or in other words, the parcel of the kinetic energy directed to translational and rotational energy?

Is it possible to calculate the minimum inclination of the ramp that satisfies the statement above?

If needed use other variables like h for the height of the cylinder, g for gravity, etc...
 
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  • #2
jaumzaum said:
If we had a cylinder rolling down a ramp, with scalar friction coefficient us and kinetic friction coefficient uc, and we assume for example that the inclination of the ramp is enough to make the cylinder reach the acceleration needed to exceed the scalar friction of rolling.
I assume you mean that the slope is sufficient that the forces overcome static friction. If the slope is constant, it's not a question of the cylinder reaching such a condition after some time; it'll happen straight away or not at all.
Suppose cylinder has mass m, moment of inertia I (you didn't say whether it was hollow or solid, etc.) linear acceleration a, radius r. The normal force is N, the frictional force F. Slope has angle θ to vertical.
Forces parallel to the slope: ma = mg cos θ - F
Perpr. to slope: N = mg sin θ
Angular accn = a/r
Moments about cylinder's centre: Fr = Ia/r

At rolling limit: F = usN = us mg sin θ
Fr2 = us mgr2 sin θ = Ia = mgr2 cos θ - mar2
a = mgr2 cos θ /(I + mr2)
tan θ = I /((I + mr2)us)

Once it has started to slip, F reduces to ucN, so the acceleration will increase a bit.
 

Related to How Does Cylinder Dynamics Change When Rolling Down a Steep Inclined Plane?

1. What is the concept of a cylinder rolling down a ramp?

The concept of a cylinder rolling down a ramp is an example of rotational motion, where the cylinder experiences both translational and rotational motion as it moves down the ramp. The ramp provides an inclined surface for the cylinder to roll on, and the force of gravity causes the cylinder to accelerate down the ramp.

2. What factors affect the speed of a cylinder rolling down a ramp?

The speed of a cylinder rolling down a ramp is affected by several factors, such as the height of the ramp, the angle of incline, the mass and diameter of the cylinder, and the surface of the ramp. The steeper the ramp, the faster the cylinder will roll. A heavier and larger cylinder will also accelerate faster than a lighter and smaller cylinder.

3. How do you calculate the velocity of a cylinder rolling down a ramp?

The velocity of a cylinder rolling down a ramp can be calculated using the formula v = √(2gh), where v is the velocity, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the ramp. This formula assumes that there is no friction or air resistance acting on the cylinder.

4. What is the significance of the shape of the ramp in a cylinder rolling down?

The shape of the ramp is significant in a cylinder rolling down because it determines the angle of incline, which affects the speed and motion of the cylinder. A steeper ramp will result in a faster and more accelerated motion, while a shallower ramp will result in a slower and more controlled motion. The shape of the ramp also determines the path of the cylinder as it rolls down.

5. How does the mass of the cylinder affect its motion down a ramp?

The mass of the cylinder affects its motion down a ramp by influencing its acceleration. According to Newton's Second Law of Motion, F = ma, where F is the force, m is the mass, and a is the acceleration. A heavier cylinder will require a greater force to accelerate down the ramp compared to a lighter cylinder, resulting in a slower motion. However, once the cylinder is in motion, its mass will not affect its speed down the ramp.

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