A solid cylinder with a radius of 0.2 meters and a mass of 2 kg rolls down an incline with an angle of 30 degrees. If the coefficient of kinetic friction is 0.1, calculate its acceleration and speed after 5 seconds.
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To calculate the acceleration, we need to find the net force acting on the cylinder.
The weight force (mg) can be decomposed into two components:
- The force parallel to the incline (mg * sinθ)
- The force perpendicular to the incline (mg * cosθ)
The force parallel to the incline is balanced by the kinetic friction force (μk * N), where N is the normal force acting on the cylinder.
The normal force (N) can be calculated as the perpendicular component of the weight force:
N = mg * cosθ
Thus, the net force (Fnet) can be written as the force parallel to the incline minus the kinetic friction force:
Fnet = mg * sinθ - μk * mg * cosθ
The acceleration (a) can be calculated using Newton's second law:
Fnet = ma
Therefore, we have:
mg * sinθ - μk * mg * cosθ = ma
Substituting the values:
mass (m) = 2 kg
acceleration due to gravity (g) = 9.8 m/s^2
angle of the incline (θ) = 30 degrees
kinetic friction coefficient (μk) = 0.1
a(2 kg)(9.8 m/s^2) * sin(30°) - (0.1)(2 kg)(9.8 m/s^2) * cos(30°) = (2 kg)a
Simplifying the equation, we find:
19.6 m * sin(30°) - 1.96 m * cos(30°) ≈ 2a
Now, let's calculate the acceleration (a):
a ≈ (19.6 m * sin(30°) - 1.96 m * cos(30°))/(2 kg)
a ≈ 2.36 m/s^2
To calculate the speed after 5 seconds, we can use the following equation of motion:
v = u + at
Where:
- v is the final velocity
- u is the initial velocity (which is 0 because the cylinder starts from rest)
- a is the acceleration
- t is the time
Substituting the known values:
v = 0 + (2.36 m/s^2)(5 s)
v ≈ 11.8 m/s
Therefore, the acceleration is approximately 2.36 m/s^2 and the speed after 5 seconds is approximately 11.8 m/s.
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