The man hanging upside down is holding a partner who weighs 699 N. Assume that the partner moves on a circle that has a radius of 3.37 m. At a swinging speed of 3.49 m/s, what force must the man apply to his partner in the straight-down position?

To find the force that the man must apply to his partner in the straight-down position, we can start by considering the relationship between centripetal force and the circular motion of the partner.

1. First, we need to calculate the centripetal force acting on the partner. The formula for centripetal force is given by F = (m * v^2) / r, where F is the centripetal force, m is the mass of the partner, v is the velocity, and r is the radius of the circle.

2. In this case, the mass of the partner is not given directly. Instead, we are given the weight of the partner, which is the force acting on it due to gravity. The weight can be calculated using the formula weight = mass * acceleration due to gravity. In this case, the weight is 699 N.

3. To find the mass, we can rearrange the weight formula to mass = weight / acceleration due to gravity. The acceleration due to gravity is approximately 9.8 m/s^2.

mass = 699 N / 9.8 m/s^2
= 71.33 kg

4. Substituting the values we have into the centripetal force formula, we have:

F = (m * v^2) / r
= (71.33 kg * (3.49 m/s)^2) / 3.37 m
= 249.84 N

So, the centripetal force acting on the partner in the circular motion is approximately 249.84 N.

5. Now, we need to find the force that the man must apply to his partner in the straight-down position. This force is equal to the weight of the partner plus the centripetal force acting on the partner.

Force = weight + centripetal force
= 699 N + 249.84 N
= 948.84 N

Therefore, the force that the man must apply to his partner in the straight-down position is approximately 948.84 N.