Calculate the force (in N) the woman in the figure below exerts to do a push-up at constant speed, taking all data to be known to three digits.

To calculate the force the woman exerts during a push-up at constant speed, we need to use principles of equilibrium and torque.

### Problem Setup:

- **Mass of the woman (\(m\))**: 53 kg
- **Distance from her feet to her center of mass (\(d_1\))**: 0.86 m
- **Total distance from her feet to her hands (\(d_2\))**: 1.43 m
- **Acceleration due to gravity (\(g\))**: 9.81 m/s²

### Step 1: Identify Forces

During a push-up, the woman exerts a force \( F_h \) through her hands and a force \( F_f \) through her feet. The sum of these forces equals the weight of the woman:

\[
F_h + F_f = m \cdot g
\]

However, we need to determine the force exerted by her hands (\( F_h \)).

### Step 2: Apply Torque Equilibrium

To find \( F_h \), we'll take torques about her feet (the point where \( F_f \) acts) to eliminate \( F_f \) from the equation.

Torque about the feet:

\[
F_h \cdot d_2 = m \cdot g \cdot d_1
\]

### Step 3: Solve for \( F_h \)

Rearrange the torque equation to solve for \( F_h \):

\[
F_h = \frac{m \cdot g \cdot d_1}{d_2}
\]

Substitute the given values:

\[
F_h = \frac{53 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 0.86 \, \text{m}}{1.43 \, \text{m}}
\]

Calculate the force:

\[
F_h = \frac{446.97 \, \text{N} \times 0.86 \, \text{m}}{1.43 \, \text{m}} \approx \frac{384.4 \, \text{N} \cdot \text{m}}{1.43 \, \text{m}} \approx 268.9 \, \text{N}
\]

### Final Answer:

The force the woman exerts through her hands during the push-up is approximately **268.9 N**.

Would you like further details on any part of the calculation?

**Questions:**
1. What assumptions are made in the torque calculation?
2. How would the force exerted by her hands change if the center of mass was closer to her hands?
3. Why do we consider torques rather than just forces to solve this problem?
4. What role does constant speed play in this analysis?
5. How would the force exerted change if her mass increased?
6. Can this method be used to calculate forces in other exercises like planks?
7. How does the concept of equilibrium help in solving this problem?
8. How does the length of her body affect the force she needs to exert?

**Tip:** When solving problems involving torques, always choose a pivot point that simplifies the calculation by eliminating one or more unknown forces.

Learn how to calculate the force exerted by a woman during a push-up exercise using principles of equilibrium and torque. This detailed solution covers the mass, distances, and gravitational acceleration involved. Suitable for college-level physics students.

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