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A body is held at rest on a smooth plane, which is inclined to the horizontal at an angle π.
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The body is released and slides down the plane under the action of its weight.
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What is the acceleration of the body in terms of the gravitational acceleration π?
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In this exercise, we want to solve for the acceleration that the body experiences.
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Weβll call that π.
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To start on our solution, letβs draw a sketch of the body at rest on the smooth plane.
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Here, we have a sketch of our body, which weβve drawn as a box on an inclined plane at angle π.
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Our body has some mass.
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We can call that mass π.
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Our first step with this diagram is to draw the forces that are acting on this body.
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We know that gravity acts on the body.
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We can call that force πΉ sub π.
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And there is also a normal force acting perpendicular to the inclined plane that acts on the body.
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Weβve called that force πΉ sub π.
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Weβre told that the inclined plane is smooth.
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Thatβs a way of saying that there is no friction between the body and the plane surface.
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So there is no frictional force involved here.
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Under the influence of these two forces, gravity and the normal force, the body will slide and accelerate down the plane.
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To find out what its acceleration is down the plane, we can set in place a pair of corded axes with π¦ pointing perpendicular to the inclined surface and π₯ pointing up the plane.
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Based on these two axes, we can separate the gravitational force πΉ sub π into π₯- and π¦-components.
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When we do that, we see that the π¦-component and the π₯-component form a right angle.
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So we have a right triangle, where those are the two legs and the magnitude of the gravitational force is the hypotenuse.
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Itβs also true that the angle at the top corner of this right triangle is equal to the angle of our inclined plane π.
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To solve for the acceleration of this body down the plane, we want to concentrate on the π₯-component of our gravity force.
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That particular component will be equal to the magnitude of the force of gravity πΉ sub π multiplied by the sine of the angle π.
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At this point, it will be helpful to recall Newtonβs second law.
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The second law tells us that the net force acting on an object equals that objectβs mass times its acceleration.
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In our example, if we focus solely in the π₯-direction, there is only one force at play.
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Itβs the π₯-component of the gravitational force πΉ sub π times the sine of π.
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By the second law, we can say that this force is equal to the bodyβs mass times π β its acceleration.
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We can rewrite πΉ sub π, which represents the gravitational force, as the product of the bodyβs mass times the acceleration due to gravity π.
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Written this way, we see that the mass of the body cancels out from both sides of the equation.
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Our result is independent of its mas.
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And this leads us to our answer for π.
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π is equal to π times the sine of π.
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Thatβs the acceleration of the body down the plane.