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 50° above the horizontal, and the wife's pulling force is directed 31° above the horizontal. The husband pulls with a force whose magnitude is 62 N. What is the magnitude of the pulling force exerted by his wife?

Since they do the same amount of work pulling equal distances, they both pull with the same force component along the horizontal direction of motion.

F(man)*cos 50 = F(woman)*cos31

F(woman) = F(man)*[cos50/cos31]

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To solve this problem, we will use the fact that work done is equal to the force multiplied by the displacement multiplied by the cosine of the angle between the force and displacement.

Given:
- The husband's pulling force has a magnitude of 62 N and is directed 50° above the horizontal.
- The wife's pulling force is directed 31° above the horizontal.
- Both exert a constant force and pull the wagon through the same displacement.
- Both do the same amount of work.

We need to find the magnitude of the pulling force exerted by the wife.

Let's start by calculating the work done by the husband.

The magnitude of the husband's pulling force is 62 N, and the angle between the force and displacement is 50°. We can calculate the work done by the husband using the formula:

Work done by the husband = Force × Displacement × cos(angle)

So, the work done by the husband is:
Work_husband = 62 N × Displacement × cos(50°)

Now, since the husband and wife do the same amount of work, the work done by the wife will be equal to the work done by the husband.

Work_wife = Work_husband

The angle between the force and displacement for the wife's pulling force is 31°. We can now rearrange the equation above to solve for the magnitude of the wife's pulling force.

Magnitude of wife's pulling force × Displacement × cos(31°) = 62 N × Displacement × cos(50°)

Canceling out the displacement on both sides, we get:

Magnitude of wife's pulling force × cos(31°) = 62 N × cos(50°)

Finally, we can solve for the magnitude of the wife's pulling force:

Magnitude of wife's pulling force = (62 N × cos(50°)) ÷ cos(31°)

Calculating this, we find that the magnitude of the pulling force exerted by the wife is approximately 53.38 N.

To find the magnitude of the pulling force exerted by the wife, we can use the concept of work done.

The work done by a force is given by the equation:

Work = Force * Distance * Cos(theta)

Where:
- Work is the work done by the force
- Force is the magnitude of the force
- Distance is the displacement along which the force acts
- theta is the angle between the force and the displacement

In this case, both the husband and wife do the same amount of work, so their work values will be equal.

For the husband's force:
- Force = 62 N (given)
- theta = 50° (given)
- Distance is not given, but since the displacement is the same for both, we can assume it is the same for the wife's force as well.

Using the given values for the husband's force, we can calculate the work done by him.

Now, for the wife's force:
- Force = unknown (let's call it F_wife)
- theta = 31° (given)
- Distance is the same as the husband's, but not given explicitly.

Since both the husband and wife do the same amount of work, we can equate their work values:

Work_husband = Work_wife

(62 N) * Distance * Cos(50°) = F_wife * Distance * Cos(31°)

Now, we can cancel out the Distance term on both sides of the equation:

(62 N) * Cos(50°) = F_wife * Cos(31°)

From here, we can solve for F_wife by rearranging the equation:

F_wife = (62 N) * Cos(50°) / Cos(31°)

Now, we can calculate this value:

F_wife = (62 N) * 0.64278760968 / 0.857167300702

After evaluating the expression, we find that the magnitude of the pulling force exerted by the wife is approximately 46.4 N.