Remember, the definition of work by a force is:. Finally, there is a frictional force that is parallel to the incline. Since we are dealing with a rigid object, this force actually doesn't have any displacement I know that sounds crazy.
But just look at a rolling wheel, the frictional force is at the point of contact, but this force doesn't move. Instead the wheel turns and there is a new contact point. In short, you can either have a rigid object OR work done by friction, but not both. This leaves us with the following work-energy equation. Remember that the work is zero and the disk starts at position 1 from rest and not rotating.
Now I can add to this two ideas. First, I know the expression for the moment of inertia of disk. Second, the disk is rolling and not sliding. Since the disk is rolling, the speed of the center of mass of the disk is equal to the angular speed times the radius of the disk. Putting this all together, I can solve for the velocity at the bottom. Ok, but what about the acceleration? I will assume that the object rolls down the incline with a constant acceleration. In this case, it starts from rest and ends with the final speed all the time while moving a distance s down the incline.
In the direction along the incline, I can find the acceleration:. Remember, the initial velocity was zero - that's why the v 1 term drops out. But what about the time interval? Here I can use the definition of average velocity. Oh wait. Couldn't I have just used that kinematic equation?
You know, the one that looks like this:. Yes, I could easily have used that equation instead. Also, I could make waffles in the morning using one of those box mixes. Personally, I prefer to make my waffles from scratch. I should put in the value for the final velocity from the rolling part. This gives the disk an acceleration of:. The sine of this angle will be the opposite side h divided by the hypotenuse s. That means I can rewrite the equation as:.
Can we get the acceleration of the disk without using the work-energy principle? Let's start with a force diagram of the disk as it rolls down the incline. Three forces, this should be simple - right? The disk only accelerates along the x-direction along the plane so this should be a simple problem. But no. So the force causing the acceleration is the component of the force of gravity acting down the slope F D.
The force of gravity is resolved into two components, F D parallel to the slope and F P perpendicular to the slope. F P presses the object against the slop and is balanced by the normal reaction force R. We need to calculate the resultant force acting down the slope due to the weight of the body and force of friction. The force of friction will always act opposite to the direction of the motion.
The direction of the acceleration depends on the direction of the unbalanced force. Friction and the component of weight parallel to the slope are acting in opposite directions. Notes on Acceleration Graphs. Notes on Forces. Notes on Newtons 1st Law. Notes on Newtons 2nd Law. Notes on Vectors. Notes on Velocity Graphs. Types of Forces. Notes on Inclined Planes. Projectile Motion Notes. Notes on Momentum and Impulse. Circular Motion. Universal Law of Gravitation.
Energy in a Closed System. Notes on Springs. Energy in an Open System. Sound Waves Notes. Light Waves Notes. Series Circuits. Parallel Circuits. Standard Model. Static Electricity. Electrical Fields. Review Book. Another discovery Galileo made with.
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Also I have substituted a revised scenario process for the original to incorporate improvements that have evolved over the past several years of consulting at The Chasm Group.
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