A solid sphere of radius 0.2 meters rolls down an incline with a variable angle that changes with time. Determine the acceleration of the sphere as a function of time.
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To determine the acceleration of the sphere as a function of time, we need to analyze the forces acting on the sphere.
The main force acting on the sphere is the force of gravity. We can decompose the force of gravity into two components: one parallel to the incline and one perpendicular to the incline.
The perpendicular component of the gravity force does not contribute to the acceleration of the sphere, as it cancels out with the normal force exerted by the incline.
The parallel component of the gravity force causes the sphere to accelerate down the incline. This force can be expressed as:
F_parallel = m * g * sin(θ)
where m is the mass of the sphere, g is the acceleration due to gravity, and θ is the angle of the incline.
The acceleration of the sphere down the incline can be calculated using Newton's second law:
a = F_parallel / m
Since the sphere is rolling without slipping, there is a relationship between the linear acceleration and the angular acceleration:
a = R * α
where R is the radius of the sphere, and α is the angular acceleration. The angular acceleration can be expressed as:
α = a / R
Substituting the earlier equations, we have:
α = (F_parallel / m) / R
α = (m * g * sin(θ) / m) / R
α = g * sin(θ) / R
Therefore, the acceleration of the sphere as a function of time is:
a(t) = g * sin(θ(t)) / R
where θ(t) is the angle of the incline at time t.
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