Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 1.80 * 10^6 N, one 16 degrees west of north and the other 14 degrees east of north, as they pull the tanker 0.63 km toward the north. What is the total work they do on the supertanker?
I'm writing out the expression for total work: W = 2 * 1.8 * 10^6 * 630
...but how do I figure in the angles?
Thanks.
Only use the North components of the forces. Work in is the Force times the motion IN THE DIRECTION OF THE FORCE.
Fn = 1.8 * 10^6 cos 16 + 1.8*10^6 cos 14
Well, trying to figure out the angles can be quite an uphill battle, but don't worry, I'm here to help lighten the load!
To find the total work done, we need to consider the angle between each tugboat's force and the displacement of the supertanker. Since one tugboat is 16 degrees west of north and the other is 14 degrees east of north, we can assume that the net angle between them is 16 degrees + 14 degrees = 30 degrees.
Now, the total work done is given by the formula:
W = F * d * cos(theta)
Where F is the force applied, d is the displacement, and theta is the angle between the force and displacement. In this case, each tugboat exerts a force of 1.80 * 10^6 N and the displacement is 0.63 km. The angle between the force and displacement is 30 degrees.
So, putting it all together:
W = 2 * 1.8 * 10^6 * 0.63 * cos(30)
Calculating that out will give you the total work done by the tugboats on the supertanker. Just be careful not to get too tangled up in the numbers!
To incorporate the angles into the calculation of work, we need to find the components of the force in the northward direction.
The force directed northward by the tugboat pulling 16 degrees west of north can be expressed as:
F1 = 1.80 * 10^6 N * sin(16°)
Similarly, the force directed northward by the tugboat pulling 14 degrees east of north can be expressed as:
F2 = 1.80 * 10^6 N * sin(14°)
To find the total work done, we multiply the forces by the displacement and sum them up. The displacement only has a northward component since the tugboats are pulling the tanker toward the north.
Thus, the total work is given by:
W = 2 * (F1 + F2) * 0.63 km
Before evaluating the calculation, remember to convert the displacement from km to meters since force is given in newtons.
Let's calculate the answer.
To figure in the angles, you need to take into account the components of the forces in the northward direction. The force exerted by each tugboat can be divided into two components: one in the northward direction and the other perpendicular to the northward direction.
Let's consider the first tugboat. The force it exerts can be split into a component in the northward direction and a component perpendicular to the northward direction. The northward component can be found by multiplying the magnitude of the force (1.80 * 10^6 N) by the cosine of the angle between the force and the north direction (16 degrees west of north). Similarly, the perpendicular component can be found by multiplying the magnitude of the force by the sine of the angle.
So, for the first tugboat, the northward component of the force is (1.80 * 10^6 N) * cos(16 degrees), and the perpendicular component is (1.80 * 10^6 N) * sin(16 degrees).
Now, repeat this process for the second tugboat, but with a different angle (14 degrees east of north). The northward component of the force will be (1.80 * 10^6 N) * cos(14 degrees), and the perpendicular component will be (1.80 * 10^6 N) * sin(14 degrees).
Now that you have the northward components of the forces for both tugboats, you can find the total northward force by adding them together.
Finally, calculate the work done by each tug separately by multiplying its northward force by the displacement in the northward direction (0.63 km). Then, add the work done by both tugboats to find the total work done on the supertanker.
So, the expression for total work will be:
W = (Force1_north * Displacement) + (Force2_north * Displacement)
where
Force1_north = (1.80 * 10^6 N) * cos(16 degrees west of north)
Force2_north = (1.80 * 10^6 N) * cos(14 degrees east of north)
Displacement = 0.63 km
Now you can evaluate this expression to find the total work done on the supertanker.