A husband and Wife take turns pulling their child in a wagon along a horizontal sidewalk. Each exerts a constant force and pulls the wagon through the same displacement. They do the same amount of work, but the husband's pulling force is directed 58 degrees above the horizontal, and the wife's pulling force is directed 38 degrees above the horizontal. The husband pulls with a force whose magnitude is 67N. What is the magnitude of the pulling force exerted by the wife?
We can write the work done by each person as:
W = F_horizontal * d
Since the work done by the husband and the wife is the same, we can write:
Fh_horizontal * d = Fw_horizontal * d
Where Fh_horizontal and Fw_horizontal are the horizontal components of the forces exerted by the husband and the wife, respectively. Since the distances d are the same, they cancel out, and we have:
Fh_horizontal = Fw_horizontal
Now we can find the horizontal components of the forces exerted by the husband and the wife using their angles with respect to the horizontal:
Fh_horizontal = Fh * cos(θh) = 67N * cos(58°)
Fw_horizontal = Fw * cos(θw) = Fw * cos(38°)
Now we can plug Fh_horizontal into the equation:
Fw * cos(38°) = 67N * cos(58°)
We want to find Fw, so we can rewrite this equation as:
Fw = (67N * cos(58°)) / cos(38°)
Now we can plug in the values for the angles:
Fw = (67N * cos(58°)) / cos(38°) = (67N * 0.529) / 0.788
Fw ≈ 45.1 N
So the magnitude of the pulling force exerted by the wife is approximately 45.1 N.
To find the magnitude of the pulling force exerted by the wife, we can start by understanding the concept of work and how it relates to the given information. Work is defined as the product of the force applied and the displacement through which the force is applied.
In this case, both the husband and the wife are pulling the wagon with a constant force and over the same displacement. Since they do the same amount of work, we can set up the equation:
Work by husband = Work by wife
To calculate the work done by each person, we need to know the force applied by each person and the displacement. The displacement is the same for both, so we don't need to consider it in this case.
Let's denote the force applied by the wife as Fw and the force applied by the husband as Fh.
The work done by the husband (Wh) is given by:
Wh = Fh * D * cos(θh)
Where:
Fh is the magnitude of the pulling force exerted by the husband (67N in this case),
D is the displacement, and
θh is the angle between the force and the direction of displacement for the husband (58 degrees above the horizontal in this case).
Similarly, the work done by the wife (Ww) is given by:
Ww = Fw * D * cos(θw)
Where:
Fw is the magnitude of the pulling force exerted by the wife,
θw is the angle between the force and the direction of displacement for the wife (38 degrees above the horizontal).
Since both Wh and Ww are the same, we can set up the equation:
Fh * D * cos(θh) = Fw * D * cos(θw)
The displacement D cancels out since it's common to both sides of the equation.
Now, we can plug in the known values:
67 * cos(58) = Fw * cos(38)
We can solve this equation to find the magnitude of the pulling force exerted by the wife (Fw):
Fw = (67 * cos(58)) / cos(38)
Using a calculator, this simplifies to:
Fw ≈ 78.32 N
Therefore, the magnitude of the pulling force exerted by the wife is approximately 78.32 N.