A solid cylinder with a radius of 0.2 meters and a mass of 5 kg rolls down an incline without slipping. Calculate its acceleration and the time it takes to reach the bottom of the incline.
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To calculate the acceleration of the solid cylinder, we can use the rotational dynamics equation:
τ = Iα,
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
Since the cylinder is rolling without slipping, the torque can be expressed as the product of the normal force (N) and the radius (r):
τ = N * r.
The normal force can be found by considering the forces acting on the cylinder. The gravitational force (mg) can be broken down into two components: the component parallel to the incline (mg * sin(θ)) and the component perpendicular to the incline (mg * cos(θ)). The normal force is equal in magnitude but opposite in direction to the perpendicular component of the gravitational force, so N = mg * cos(θ).
The moment of inertia of a solid cylinder rotating about its central axis is given by the equation:
I = (1/2) * m * r^2,
where m is the mass of the cylinder and r is the radius of the cylinder.
Substituting these equations into the rotational dynamics equation, we have:
N * r = (1/2) * m * r^2 * α.
The mass of the cylinder cancels out, and we can solve for α:
α = (2 * N) / r.
The acceleration of the solid cylinder can be found by considering the forces acting on the cylinder along the incline:
ma = mg * sin(θ),
where m is the mass of the cylinder, a is the linear acceleration, and θ is the angle of the incline. Note that the torque, τ, is equal to the product of the moment of inertia, I, and the angular acceleration, α:
τ = I * α.
Since the cylinder is rolling without slipping, the linear acceleration, a, can be related to the angular acceleration, α, by the equation:
a = r * α.
Substituting the expression for α derived previously, we have:
a = (2 * N * r) / (r * m) = (2 * N) / m.
Now, we can substitute the expression for N:
a = (2 * mg * cos(θ)) / m = 2 * g * cos(θ).
Given that the radius of the cylinder is 0.2 meters and the mass is 5 kg, we can calculate the acceleration for a given angle of inclination, θ. The acceleration is independent of the radius and mass in this case:
a = 2 * 9.8 m/s^2 * cos(θ) = 19.6 m/s^2 * cos(θ).
To calculate the time it takes for the cylinder to reach the bottom of the incline, we can use the kinematic equation:
s = 0.5 * a * t^2,
where s is the distance traveled along the incline and t is the time taken.
The distance traveled along the incline can be found using the arc length formula:
s = r * θ,
where r is the radius of the cylinder and θ is the angle of inclination.
Substituting this expression for s into the kinematic equation, we have:
r * θ = 0.5 * a * t^2.
Solving for t, we find:
t^2 = (2 * r * θ) / a,
t = sqrt((2 * r * θ) / a).
Given the radius is 0.2 meters and θ is the angle of inclination, we can calculate the time it takes for the cylinder to reach the bottom of the incline using the derived acceleration.
Note that if θ = 0, then the cylinder is on a flat surface, and the time would be zero.
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