Derive an expression for acceleration for pure rolling down an inclined plane?
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Question: Derive an expression for acceleration for pure rolling down an inclined plane?
When an object rolls without slipping down an inclined plane, it experiences both translational and rotational motion. The acceleration of the object can be calculated by considering the forces acting on it. The force of gravity acts on the object, causing it to accelerate down the plane, while the frictional force opposes the motion and causes a torque that affects the rotational motion of the object.
The acceleration of the object can be calculated using the following expression:
a = (g sinθ)/(1 + (I/(mr^2)))
where a is the acceleration of the object, g is the acceleration due to gravity, θ is the angle of the inclined plane, I is the moment of inertia of the object, m is its mass, and r is the radius of the object.
This expression shows that the acceleration of the object is dependent on the angle of the plane, the moment of inertia of the object, and the radius of the object. As the moment of inertia of the object or the radius of the object increases, the acceleration decreases. Conversely, as the angle of the plane increases, the acceleration increases.
Understanding the acceleration of objects that roll down inclined planes is important in many practical applications, such as in the design of roller coasters and the analysis of the motion of vehicles on steep roads.
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