In order to paint the outside of your house, you buy a 24 ft aluminum extension ladder. The ladder weighs 61 pounds, and has rubber caps on the bottom of the legs. You put the base of the ladder on the sidewalk 7.5 feet from the side of your house. Can you safely climb all the way up the ladder? Assume that there is no friction with the side of the house.
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In order for you to climb the ladder safely, it must remain in place as you climb. In other words, you want the motion of the ladder to remain at rest both translationally and rotationally as the location of the forces on the ladder changes. Any time you compare forces and motion of an object, it is best understood through Newton's Second Law. Because you need to consider rotational as well as translational motion, you will need to work with the rotational equivalent of the law as well as with the linear version.
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When we work with translational motion, we picture the action of forces with a free body diagram. When we work with rotational motion, the location of the forces also matters. Therefore, we need both a free body diagram and a picture showing the forces at the points where they act. If your drawing gets too cluttered, you can lable the geometric quantities on yet another drawing as shown below.
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The key equation for any problem that relates forces and motion is Newton's Second Law. The left side of the equation takes into account the forces that act on the object, and the right side shows the effect of those forces. To examine rotational effect, we look at the torques on the object. To examine translational effect, we look at the forces on the object.
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It does not matter if you begin this problem by examining force or torque. Past experience has shown me that it is often true that the force equations can be solved and the results used in the torque equation. However, beginning with torque is equally fine--scroll down to see that part of the solution.
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Step 2
At this angle, and with these surfaces, you can safely climb a distance much longer than the length of the ladder. So yes, you can safely climb it.
The angles of a triangle add to 1800. The two known angles in this triangle are 900 and 720. Therefore, the remaining angle (the angle that a vertical line makes with the ladder) is 180.
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In order for you to climb a ladder safely, the ladder needs to remain in place. There must not be any translational or rotational motion. This means that the forces on the ladder need to balance and also that the torques on the ladder need to balance. In this case, the ladder is on a rough surface and put fairly steeply in against the wall--it makes sense that forces and torques will balance.
You may also have noticed that very little information was provided. Be confident in your approach to the physics by clearly articulating why you approach the problem as you do. If you are confident that you are looking at the appropriate interactions, you can look up tabulated data that you need.
Finally, note that there are several ways to approach the problem. You can either use the maximum available friction to find how high you can climb, or you can find the friction needed to climb the length of the ladder. Both approaches will lead to the same conclusion.
Because your solution is independent of mass, it is not limited to only heavy vehicles like trucks.
(58 ft/s)(1 mi/5280 ft)(3600 s/hr) = 39 mi/hr