A force of 18.0 lb is required to hold a 60.0-lb stump grinder...

Question

A force of 18.0 lb is required to hold a 60.0-lb stump grinder on an incline. What angle does the incline make with the horizontal?

Accepted Answer

Hey, everyone. Here we are given a person is holding an £85 trolley bag on the ramp outside an airport, find the angle of the ramp with the ground. If the bag experiences a force of £29 we have four answer choice options. A 18 degrees B 30 degrees C 19 degrees and D 20 degrees. So to begin this problem, we need to start by creating a sketch containing all of our given information. So to start our sketch, we just want to draw a horizontal line to represent our ground. And now we need to draw in the ramp. So we need to add another line coming upwards from the ground at some angle. And now we want to label the angle in between the ramp and the ground as angle theta and this is the angle we are trying to find. Next. We can represent our bag by drawing a rectangle on our ramp. And now we are told that the bag experiences a force of £29. So we can represent this force as an arrow going from the rectangle upwards along the ramp. And now we just need to label this arrow as the £29 force. Lastly, we are told that the, the bag weighs £85. So this tells us we have a weight force of £85. And so to illustrate the weight force, we need to draw another arrow going from the rectangle directly downwards. And this arrow must be perpendicular to the ground. And now we will label this arrow as £85. And so now we have finished sketching all of our given information. So we want to begin finding theta. So we first want to complete our triangle that we have forming underneath our rectangle. So we just need to draw a line coming down from our ramp to the ground. And this line must be perpendicular to the ground. And thus, we have our 90 degree angle formed. However, looking at our triangle, we can notice we do not have any of these sides values known. So we cannot find our value for the. So instead we want to recreate this triangle utilizing our weight force. And to do this, we just need to draw another line going from the rectangle downwards. However, this line must be perpendicular to the ramp. And now, since we have a line perpendicular to the ramp and our weight force is perpendicular to the ground, this tells us that the angle in between these two lines is our angle theta. And now to complete our triangle, we notice that our £29 force is parallel to the ramp. So that means it is perpendicular to one of these sides of our triangle. So to finish the triangle, we just need to draw our £29 force going from our newly added line to the weight force so that the heads of both arrows are touching. And so we have formed our 90 degree angle and we just need to label this arrow as £29. And with this, we have finished our new triangle where we can notice we have one side value nome. And we also have the hypotenuse nome. So we just want to recall that sign is equivalent to the opposite side from the angle divided by the hypotenuse. And so now with all of this information, we can easily find our value for theta by noticing that sign of theta is equivalent to £29 divided by £85. And now just solving for the, we can first notice that the pound units cancel each other out. And so now we just want to take the sine inverse of both sides. And so we have that theta is equal to sine inverse of 29 divided by 85. Now just evaluating this expression sine inverse of 29 divided by 85 gives us 19.949 degrees. And now our last step here is to round our value to the nearest whole degree. And so we find that theta is approximately equal to 20 degrees. And with this final value, we are left with answer choice D where again, we found that the angle in between the ramp and the ground is 20 degrees. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

A force of 18.0 lb is required to hold a 60.0-lb stump grinder on an incline. What angle does the incline make with the horizontal?

Key Concepts

Forces on an Incline

Trigonometric Functions

Equilibrium Conditions

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