Video: Finding the Magnitude of a Smooth Inclined Plane’s Reaction to a Body Placed on It

A body of mass 0.7 kg was placed on a smooth plane inclined at 66° to the horizontal and it was left to move freely under the effect of gravity, where the acceleration due to gravity is 9.8 m/s². Find, to the nearest two decimal places, the magnitude of the reaction of the plane to the body.

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Video Transcript

A body of mass 0.7 kilograms was placed on a smooth plane inclined at 66 degrees to the horizontal and it was left to move freely under the effect of gravity, where the acceleration due to gravity is 9.8 meters per square second. Find to the nearest two decimal places the magnitude of the reaction of the plane to the body.

When looking at motion of a body on an inclined plane, it’s really helpful to draw a diagram. Our plane is inclined at 66 degrees to the horizontal. And we have a body whose mass is 0.7 kilograms on this plane. Using the fact that force is equal to mass times acceleration, we can say that the downward force of the body is equal to 0.7 times its acceleration due to gravity 𝑔. Now, actually, when we’re dealing with inclines, we’re more concerned with the motion parallel or perpendicular to the surface of the incline. And so, we will at some point need to consider the perpendicular components of the force of gravity on the mass.

Now, using Newton’s third law of motion, we know the reaction of the plane to the body will be equal an opposite to this force. So if we calculate the perpendicular component of the force for gravity, we’ll know the reaction force. So what we do is add a right-angled triangle to our diagram. We know we have a right angle here and a second one here. So we can use the fact that angles in a triangle sum to 180 degrees to calculate this angle to be 24. We then subtract 24 from 90 degrees. And we find that the angle we need in our right-angled triangle is 66 degrees.

And so, we have a right-angled triangle. We have an included angle of 66 degrees and our hypotenuse worth 0.7𝑔. We can use right angle trigonometry to calculate the value of 𝑥. We label the triangle as shown. And we see we’re looking to find the adjacent. And we know the hypotenuse: cos of 𝜃 is equal to adjacent over hypotenuse. So in this case, we can say that cos of 66 must be equal to 𝑥 over 0.7𝑔. But of course, we’re told that the acceleration due to gravity is the 9.8. 𝑔 is 9.8. So this becomes cos of 66 equals 𝑥 over 0.7 times 9.8.

And then, we can solve this equation for 𝑥 by multiplying through by 0.7 times 9.8. That tells us that 𝑥 is equal to 2.79 and so on, which gives us a force of 2.79 newtons correct to two decimal places. Since we said the reaction force 𝑅 will be equal an opposite to the perpendicular component for the force for gravity, we find 𝑅 is approximately equal to 2.79 newtons.

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