Question Video: Finding the Angle of Static Friction between a Body on an Inclined Rough Plane and the Plane Mathematics

The figure shows a body on a rough inclined plane. Given that it is on the point of sliding down the plane, find the measure of the angle of static friction between the body and the plane rounding your answer to the nearest minute if necessary.

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Video Transcript

The figure shows a body on a rough inclined plane. Given that it is on the point of sliding down the plane, find the measure of the angle of static friction between the body and the plane rounding your answer to the nearest minute if necessary.

In order to answer this question, we need to know what we mean when we talk about the angle of static friction between the body and the plane. By recalling that the coefficient of friction is the tangent of the angle at which the object slides, then we say that the angle of static friction is in fact the measure of this angle. Weโ€™re told that our object is indeed on the point of sliding down the plane. So defining the angle between the plane and the ground to be ๐œƒ, then the coefficient of friction is the tan of this angle ๐œƒ. Then the angle ๐œƒ is itself the angle of static friction.

So how do we calculate the value of ๐œƒ? Well, we might notice that we can form a right triangle including this angle ๐œƒ. The side opposite this right triangle weโ€™re given to be 23 centimeters, and the side adjacent to it is 76 centimeters. So we can in fact use the tangent ratio to find the value of ๐œƒ. Since tan of ๐œƒ is opposite over adjacent, in this case, tan of ๐œƒ is 23 over 76. This is the value of ๐œ‡; itโ€™s the coefficient of friction between the body and the plane. But weโ€™re not quite done. We want to find the angle of static friction and thatโ€™s the value of ๐œƒ. So to solve this equation, weโ€™ll take the inverse tan of both sides. Then ๐œƒ is the inverse tan or arctan of 23 over 76. Thatโ€™s 16.837 degrees and so on.

Now, the question tells us to give our answer to the nearest minute. So we take the decimal part 0.837 and we multiply that by 60. Thatโ€™s 50.24 and so on, which, correct to the nearest minute, is 50. So the angle of static friction between the body and the plane itself is 16 degrees and 50 minutes.

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