A 17.0 kg crate is pulled a distance of 20.0 m requiring that 1210 Joules of work be done. If the job is done by attaching a rope and pulling with a force of 75 Newtons, at what angle is the rope held?
work = force * distance
1210 = f * 20.0 ... f = 60.5 N
60.5 N = 75 N * cos(Θ)
Well, let me do some calculations while juggling some numbers for you. We know that work is equal to force multiplied by distance, so W = F * d. Plugging in the values we have: 1210 J = 75 N * 20.0 m.
Now, to determine at what angle is the rope held, we need to use some trigonometry. The equation we'll be using is: cosine(theta) = adjacent/hypotenuse. In this case, the adjacent side is the force applied (75 N), and the hypotenuse is the force of gravity acting on the crate (mass * gravity).
Let's calculate the force of gravity first: force of gravity = mass * gravity = 17.0 kg * 9.8 m/s^2.
Now we can calculate the angle theta: cosine(theta) = adjacent/hypotenuse, cosine(theta) = 75 N / (17.0 kg * 9.8 m/s^2). Taking the inverse cosine of both sides will give us the angle.
So, after a quick juggling of numbers, we get theta ≈ 55.7 degrees.
Therefore, the rope is held at an angle of approximately 55.7 degrees.
To find the angle at which the rope is held, we need to use the concept of work done and the dot product of force and displacement.
The work done (W) can be calculated using the formula:
W = F * d * cosθ
Where:
W is the work done (given as 1210 Joules)
F is the force applied (given as 75 Newtons)
d is the displacement (given as 20.0 m)
θ is the angle between the force and displacement (to be found)
Rearranging the formula, we get:
cosθ = W / (F * d)
Now, let's substitute the given values into the equation:
cosθ = 1210 J / (75 N * 20.0 m)
cosθ = 1210 J / 1500 N*m
cosθ ≈ 0.8067
To find the angle θ, we need to take the inverse cosine (cos^-1) of 0.8067:
θ ≈ cos^-1(0.8067)
Using a calculator, we find that θ ≈ 38.73°
Therefore, the rope is held at an angle of approximately 38.73°.
To find the angle at which the rope is held, we can use the work-energy theorem. The work done on an object is equal to the change in its kinetic energy.
The formula to calculate work is given by:
Work (W) = Force (F) * Displacement (d) * cos(theta)
In this case, the force exerted on the crate is 75 Newtons and the displacement is 20.0 meters. The work done is given as 1210 Joules.
Substituting these values into the formula, we have:
1210 J = 75 N * 20.0 m * cos(theta)
Now, let's isolate the angle theta by rearranging the formula:
cos(theta) = 1210 J / (75 N * 20.0 m)
cos(theta) = 1.613
To find the angle, we need to take the inverse cosine (cos^(-1)) of both sides of the equation:
theta = cos^(-1)(1.613)
Using a scientific calculator, we find:
theta ≈ 25.97 degrees
Therefore, the rope is held at an angle of approximately 25.97 degrees.