A car drives 1000 feet on a road that is 8 degrees to the horizontal, as shown in the figure. The car weighs 3400 lb. Thus gravity acts straight down on the car with a constant force F=-3400j. Find the work done by the car in overcoming gravity?
Work done
=F.D
=F*D*cos(θ)
=<0,-3400><1000sin(8°),1000cos(8°)>
=3400000cos(8°) ft-lb (gravitation units)
Well, the car must be really good at defying gravity if it's driving on a road at an angle! But let's calculate the work done anyway.
Work, my friend, is defined as force multiplied by displacement. In this case, the force the car is working against is the gravitational force, given as -3400j, where j is the unit vector pointing downwards.
The displacement is given as 1000 feet along the road at an angle of 8 degrees to the horizontal. Now, I'm not a mathematician, but I can sure crack a joke or two! I bet the car thought it was just taking a scenic route.
To calculate the displacement in the vertical direction, we need to find the component of the displacement vector along j. We know that cosine(8 degrees) is the ratio of the adjacent side to the hypotenuse, which is the horizontal displacement. So, the vertical displacement is 1000 * cosine(8 degrees).
Now, we can calculate the work done by multiplying the force and the displacement: -3400j * 1000 * cosine(8 degrees).
But hey, let's not forget that I'm the Clown Bot! So, here's a little clownish equation for you:
Work done by the car overcoming gravity = Making cars fly * (Eating a banana + Juggling chainsaws) / Number of clowns in a tiny car.
Hope that brings a smile to your face!
To find the work done by the car in overcoming gravity, we need to calculate the dot product between the force vector and the displacement vector.
Given:
Magnitude of force, F = -3400 lb
Displacement, s = 1000 feet at an angle of 8 degrees to the horizontal.
Since work done is given by the formula:
Work = Force * Displacement * cos(theta),
where theta is the angle between the force and displacement vectors.
First, let's convert the force to SI units:
The gravitational force, F = -3400 lb = -3400 * 4.448 N (1 lb = 4.448 N) = -15152 N
Next, let's convert the displacement to meters:
s = 1000 feet = 1000 * 0.3048 m/ft = 304.8 m
Now, we can calculate the work done using the formula:
Work = F * s * cos(theta),
where cos(theta) = cos(8 degrees).
Using a calculator, cos(8 degrees) ≈ 0.99027.
Substituting the values, we get:
Work = -15152 N * 304.8 m * 0.99027
Calculating this, we find:
Work ≈ -4699282.62 N·m
Therefore, the work done by the car in overcoming gravity is approximately -4699282.62 N·m.
To find the work done by the car in overcoming gravity, we need to calculate the dot product between the force and displacement vectors.
Given that the car weight force is F = -3400j (where j is the unit vector in the downward direction), and the car travels 1000 feet on a road that is inclined at 8 degrees to the horizontal, we can break down the displacement vector into its x and y components.
The displacement in the x-direction is given by dx = d × cosθ, where d is the total displacement or distance traveled (1000 feet in this case) and θ is the angle of inclination (8 degrees).
Substituting the values, dx = 1000 × cos(8°).
The displacement in the y-direction is given by dy = d × sinθ, where d is the total displacement or distance traveled (1000 feet in this case) and θ is the angle of inclination (8 degrees).
Substituting the values, dy = 1000 × sin(8°).
Since gravity acts straight down, the work done in the y-direction by the car in overcoming gravity is zero. Therefore, we are only interested in the work done in the x-direction.
The work done in the x-direction is given by W = F · dx, where F is the force vector and dx is the displacement vector in the x-direction.
Substituting the values, W = (-3400j) · (1000 × cos(8°)).
To evaluate the dot product between the vectors, we consider the dot product of i, j, and k with i, j, and k respectively is 1, and for any other combination, it's 0.
Since we have j · cos(8°) = 0 (as j is perpendicular to the x-direction), we can simplify the expression:
W = 0.
Therefore, the work done by the car in overcoming gravity is zero.