An 845-N sled is pulled a distance of 190 m. The task requires 1.70 104 J of work and is done by pulling on a rope with a force of 124 N. At what angle is the rope held?
We can start by finding the work done using the formula: work = force × distance × cos(θ).
The work done is given as 1.70 × 10^4 J, and the distance is 190 m, and the force is 124 N. We can plug these values into the formula above, then solve for the angle θ:
1.70 × 10^4 J = 124 N × 190 m × cos(θ)
To isolate the cos(θ), we can divide both sides of the equation by (124 N × 190 m):
cos(θ) = (1.70 × 10^4 J) / (124 N × 190 m)
Now we can calculate the value of cos(θ):
cos(θ) ≈ 0.719
And finally, we can find the angle θ by taking the inverse cosine of 0.719:
θ ≈ cos^(-1)(0.719) ≈ 44.35 degrees
So, the rope is held at an angle of approximately 44.35 degrees.
To find the angle at which the rope is held, we can use the work formula:
Work = Force * Distance * Cosine(angle)
Given:
- Work (W) = 1.70 * 10^4 J
- Force (F) = 124 N
- Distance (d) = 190 m
We need to rearrange the formula to solve for the angle (θ):
Cosine(angle) = Work / (Force * Distance)
Angle (θ) = ArcCos( Work / (Force * Distance) )
Let's substitute the given values and calculate the angle:
Angle (θ) = ArcCos( 1.70 * 10^4 J / (124 N * 190 m) )
Angle (θ) ≈ ArcCos( 0.114984...)
Using a calculator or trigonometric table, we can find the arc cosine of 0.114984...:
Angle (θ) ≈ 84.68 degrees (rounded to two decimal places)
Therefore, the rope is held at an angle of approximately 84.68 degrees.
To find the angle at which the rope is held, we can use the equation:
Work = Force * Distance * cos(theta)
where Work is the work done (1.70 * 10^4 J), Force is the magnitude of the force applied to the sled (124 N), Distance is the distance the sled is pulled (190 m), and theta is the angle between the force and the direction of motion.
Rearranging the equation, we have:
cos(theta) = Work / (Force * Distance)
Substituting the given values, we get:
cos(theta) = (1.70 * 10^4 J) / (124 N * 190 m)
Now, let's solve for cos(theta):
cos(theta) = (1.70 * 10^4 J) / (124 N * 190 m)
≈ 0.071875
To find the angle theta, we can use the inverse cosine function (cos^(-1)):
theta = cos^(-1)(0.071875)
Evaluating this on a calculator, we find:
theta ≈ 85.14 degrees
Therefore, the angle at which the rope is held is approximately 85.14 degrees.