A boy pulls a sled with a force of 40N on a rope 2.5 meters long. The end that he holds is 1.5 meters higher than the end attached to the sled. What is the magnitude of the horizontal component that acts to pull the sled forward?
Answer
To find the magnitude of the horizontal component that acts to pull the sled forward, we can use trigonometry.
First, let's visualize the situation. We have a right-angled triangle formed by the rope, with the vertical distance of 1.5 meters and the horizontal distance of 2.5 meters. The force of 40N is being applied at an angle between the vertical and horizontal directions.
The horizontal component of the force is what we are trying to find. Let's call it Fx.
To find Fx, we can use the trigonometric function cosine. The cosine of an angle is equal to the adjacent side divided by the hypotenuse. In this case, the adjacent side is Fx, and the hypotenuse is the force of 40N.
Therefore, we have:
cos(angle) = Fx / 40N
Now, we need to find the angle between the vertical and horizontal directions. In this case, that angle can be found using the opposite side and the adjacent side.
The opposite side represents the vertical distance of 1.5 meters, and the adjacent side is the horizontal distance of 2.5 meters. To find the angle, we can use the trigonometric function tangent:
tan(angle) = opposite / adjacent
In this case, we have:
tan(angle) = 1.5m / 2.5m
Now, we can solve for the angle using the inverse tangent function (often denoted as arctan or atan):
angle = atan(tan(angle))
Once we have the angle, we can substitute it back into the equation for cosine:
cos(angle) = Fx / 40N
Now we can solve for Fx by rearranging the equation:
Fx = cos(angle) * 40N
By substituting the calculated angle into the equation, we can find the magnitude of the horizontal component that acts to pull the sled forward.