WEBVTT
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A body slides down a smooth plane under the action of its weight.
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Which one of the following variables does the acceleration of the body depend on?
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Is it (A) the reaction of the plane, (B) the weight of the body, (C) the mass of the body, or (D) the angle of inclination of the plane?
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Letβs begin by sketching a diagram to model the scenario.
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If we let the mass of the body be π kilograms, then its weight exerts a force vertically downwards equal to mass multiplied by gravity.
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The weight π is equal to ππ.
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We have a normal reaction force π
acting perpendicular to the plane.
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And we will let the angle of inclination to the plane be π degrees.
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As the body is sliding down the plane under the action of its weight, there are no extra forces we need to consider.
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And as the plane is smooth, there will be no frictional force.
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We need to calculate the components of the weight force that are parallel and perpendicular to the plane.
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Using our knowledge of right angle trigonometry, the force perpendicular to the plane is equal to ππ cos π and the force parallel to the plane is equal to ππ sin π.
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Newtonβs second law states that the sum of the forces is equal to the mass multiplied by acceleration.
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Resolving perpendicular to the plane, the sum of our forces is equal to π
minus ππ multiplied by cos π.
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As the body is not accelerating in this direction, this is equal to zero.
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And the normal reaction force π
is therefore equal to ππ multiplied by cos π.
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We can therefore conclude that the reaction of the plane does not impact the acceleration of the body.
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Letβs now consider what happens when we resolve parallel to the plane.
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The only force acting on the body in this direction is ππ multiplied by sin π.
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This is equal to the mass π multiplied by the acceleration π.
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We can divide both sides of this equation by the mass π.
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This means that the acceleration π is equal to π sin π.
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And we can therefore conclude that neither the mass nor the weight of the body impacts the acceleration.
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The only variable that will affect the acceleration of the body is the angle of inclination π.
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We can therefore conclude that the correct answer is option (D).
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The acceleration of the body depends on the angle of inclination of the plane.
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We can actually go one stage further.
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As π lies between zero and 90 degrees, we know that as the angle π increases, the acceleration increases.
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This is because the function sin π is increasing between zero and 90 degrees.