A sled is dragged along a horizontal path at a constant speed of v𝑣 = 1.56 m/s by a rope that is inclined at an angle of θ𝜃 = 31.5° with respect to the horizontal (as shown in the figure below). The total weight of the sled is 488 N. The tension in the rope is 244 N. How much work is done by the rope on the sled in a time interval of 11.2 s?
Question
A sled is dragged along a horizontal path at a constant speed of v𝑣 = 1.56 m/s by a rope that is inclined at an angle of θ𝜃 = 31.5° with respect to the horizontal (as shown in the figure below). The total weight of the sled is 488 N. The tension in the rope is 244 N. How much work is done by the rope on the sled in a time interval of 11.2 s?
Solution
To solve this problem, we need to use the formula for work done, which is:
Work = Force x Distance
We know the force (which is the tension in the rope), but we need to find the distance the sled has moved in the given time. We can find this using the formula:
Distance = Speed x Time
We know the speed (v = 1.56 m/s) and the time (t = 11.2 s), so we can calculate the distance:
Distance = 1.56 m/s x 11.2 s = 17.472 m
However, the force from the rope is not entirely in the direction of the sled's motion. It is inclined at an angle of 31.5° to the horizontal. So, we need to find the component of the force that is in the direction of the sled's motion. We can do this using trigonometry:
Horizontal component of force = Force x cos(θ) = 244 N x cos(31.5°) = 244 N x 0.848 = 207.072 N
Now we can calculate the work done:
Work = Force x Distance = 207.072 N x 17.472 m = 3618.7 J
So, the work done by the rope on the sled in a time interval of 11.2 s is approximately 3618.7 Joules.
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