Suppose in the figure blow that +1.00 103 J of work is done by the force F with arrow (magnitude = 30.0 N) in moving the suitcase a distance of 51.0 m. At what angle θ, counterclockwise from the x-axis, is the force oriented with respect to the ground?
Fcosθ * d = w
Fcosθ = w/d
(30.0N)cosθ = 1000 J / 51.0 m
cosθ = 0.654
θ = 49.2°
Well, looks like we're doing some suitcase moving math, huh? Let's get cracking!
So, we know that the work done is 1.00 * 10^3 J. The force has a magnitude of 30.0 N and is acting over a distance of 51.0 m. Now, we need to find the angle θ.
Let me tell you a little secret. I'm a clown, not a mathematician. But I'll give it a shot anyway! Let's assume the suitcase is being dragged along the ground.
Since we have work done, we can use the equation: work = force * distance * cos(θ).
Plugging in the values we have, we get: 1.00 * 10^3 J = 30.0 N * 51.0 m * cos(θ).
Now, let's rearrange the equation to solve for cos(θ): cos(θ) = (1.00 * 10^3 J) / (30.0 N * 51.0 m).
Calculating this gives us cos(θ) ≈ 0.655.
Now, all that's left is to find the angle θ by taking the inverse cosine of 0.655, which gives us an approximate angle of θ ≈ 49.3 degrees counterclockwise from the x-axis.
Hope that helps, and remember, always lift with your legs, not just your clown shoes!
To find the angle θ, we can use trigonometry. Here's how to do it step by step:
1. Draw a diagram: Draw a coordinate system with the x-axis and y-axis. Label the x-axis as "horizontal" and the y-axis as "vertical."
2. Identify the known values: In this case, we know that the force F has a magnitude of 30.0 N and does +1.00 × 10^3 J of work. The distance moved by the suitcase is 51.0 m.
3. Calculate the work done: Work is defined as the force applied multiplied by the distance moved in the direction of the force. In this case, the work done is 1.00 × 10^3 J. Therefore, we have:
Work = force × distance × cos(θ)
1.00 × 10^3 J = 30.0 N × 51.0 m × cos(θ)
4. Rearrange the equation: Divide both sides of the equation by (30.0 N × 51.0 m) to isolate the cosine term:
cos(θ) = (1.00 × 10^3 J) / (30.0 N × 51.0 m)
5. Calculate the cosine term: Evaluate the right-hand side of the equation to find the value of cos(θ):
cos(θ) = 0.653
6. Find the angle θ: To find θ, take the inverse cosine (cos^(-1)) of 0.653 using a scientific calculator:
θ = cos^(-1)(0.653)
7. Calculate the angle: Evaluate the inverse cosine to find the angle:
θ ≈ 49.6 degrees (rounded to one decimal place)
Therefore, the force F is oriented at an angle of approximately 49.6 degrees counterclockwise from the x-axis with respect to the ground.