WEBVTT
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A point mass π is on a slope, as shown in the diagram.
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The weight of the mass, π, and the normal reaction force, π
, act on the point mass.
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What is the relationship between π
and π if the angle of the slope above the horizontal is zero?
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What is the magnitude of π
if the angle of the slope above the horizontal is 90 degrees?
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In the first question, it asked us to find the relationship between π
and π, where π
is the normal reaction force and π is the weight of the object when the angle of the slope above the horizontal is zero.
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Letβs redraw our diagram where we have an angle of zero.
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If the angle is zero degrees, that implies that we have a horizontal surface.
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Applying Newtonβs second law to the situation using only the vertical forces as those are the only forces acting on our point, we get that π
, the normal reaction force, minus π, the weight, is equal to zero, as our point mass is not accelerating up and down.
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We use the common convention that up is positive, thatβs why that π
has a positive value, and that down is negative, which is why the weight π has a negative value.
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This means the magnitude of the normal reaction force is equal to the magnitude of the weight.
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Therefore, the relationship between π
and π is that they are equal.
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For the second question, we are asked to find the magnitude of π
if the angle of the slope above the horizontal is 90 degrees.
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Letβs once again draw our diagram with an angle of 90 degrees this time.
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To solve for this question, letβs find a relationship for π
in terms of π.
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If we align our coordinate system to be parallel and perpendicular to the plane, then we must break down our weight vector into parallel and perpendicular components.
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We can see the normal reaction force will have the same magnitude as the perpendicular component of the weight.
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To find the perpendicular component of the weight, we can use trigonometry as the perpendicular component is the adjacent side of a right triangle to the angle π and the weight π is the hypotenuse of the triangle.
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So we can use the equation the cos of the angle π is equal to the adjacent side of a triangle divided by the hypotenuse of the triangle.
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Plugging in our values, we have cos π is equal to the perpendicular component of the weight divided by the weight.
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To isolate the perpendicular component of the weight, we multiply both sides by π, canceling out the π on the right side of the equation, leaving us with the weight of the object times the cos of the angle π is equal to the perpendicular component of the weight.
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We said earlier that the magnitude of π
will be equal to the magnitude of the perpendicular component of the weight, so we can replace the perpendicular component with π
.
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We should also recall that weight is ππ, so we could replace the weight π with ππ.
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So the expression for the normal reaction force is ππ cos π.
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The cos of 90 degrees is zero, which means that the normal reaction force of point mass π when the angle of the slope is 90 degrees is zero.
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The magnitude of π
as the angle of the slope above the horizontal becomes 90 degrees is zero.