Question

A man is doing push-ups. He has a mass of \(68 \mathrm{kg}\) and his center of gravity is located at a horizontal distance of \(0.70 \mathrm{m}\) from his palms and \(1.00 \mathrm{m}\) from his feet. Find the forces exerted by the floor on his palms and feet.

Short answer

Palms: 392 N, Feet: 274.4 N

Step-by-step solution

Understand the Problem

We need to find the forces exerted by the floor on the man's palms and feet while he is in a push-up position. We'll solve this by using the concept of torque equilibrium about any point. The man's weight acts at his center of mass.

Identify the Forces and Distances

The man's weight is the force due to gravity, which is given by his mass times the gravitational acceleration, \( W = mg \), where \( m = 68 \mathrm{kg} \) and \( g = 9.8 \mathrm{m/s^2} \). There are two forces exerted by the floor: \( F_p \) at the palms and \( F_f \) at the feet. The horizontal distance from the palms to the center of gravity is \(0.70\mathrm{m}\), and from the center of gravity to the feet is \(1.00\mathrm{m}\).

Calculate the Weight of the Man

The weight of the man is calculated as: \[ W = mg = 68 \times 9.8 = 666.4 \, \mathrm{N} \] This is the downward force acting at the center of gravity.

Apply Torque Equilibrium Condition

For rotational equilibrium about any point, the sum of torques must be zero. Taking torques about the feet, we calculate:\[ F_p \times 1.70 = 666.4 \times 1.00 \]Solving for \( F_p \), the force at the palms.

Solve for Force at the Palms

Rearrange the equation from Step 4 to solve for \( F_p \): \[ F_p = \frac{666.4 \times 1.00}{1.70} = 392 \mathrm{N} \]

Apply Total Force Equilibrium

The sum of vertical forces must also be zero for equilibrium. Thus, \[ F_p + F_f = 666.4 \].Substitute \( F_p = 392 \mathrm{N} \) into this equation.

Solve for Force at the Feet

Substitute the value of \( F_p \) into the equilibrium equation:\[ 392 + F_f = 666.4 \]Solving for \( F_f \), we get: \[ F_f = 666.4 - 392 = 274.4 \mathrm{N} \]

Key concepts

Torque Equilibrium

Torque equilibrium is a crucial concept used to analyze situations where objects are in a state of rotational balance. In simpler terms, an object is in torque equilibrium when the sum of all torques acting on it is zero. This means that the object is not rotating, or, if it is, it's rotating at a constant speed without any angular acceleration.

To calculate this, we need to consider:

• Each force acting on the object.
• The distance from each force to the pivot point, which is also known as the lever arm.

The torque (\( \tau \)) produced by a force can be calculated using the formula:\[ \tau = F \times r \]where \( F \) is the force applied, and \( r \) is the lever arm. In practical terms, if the sum of clockwise torques equals the sum of counterclockwise torques, the object achieves torque equilibrium. This is the principle used to determine the force on the palms in the push-up scenario.

Center of Gravity

The center of gravity of an object is the point where the total weight of the object is considered to act. For symmetric bodies, it's often located at the geometric center, but for uneven structures or distributed masses, it may differ. In the push-up problem, establishing the center of gravity is essential to determine where the weight of the man is acting.

The center of gravity plays a crucial role in solving equilibrium problems because the distribution of weight affects torque calculations. By finding the center of gravity, we help calculate how forces need to be applied to maintain balance. This concept is fundamental to the analysis of stable, unstable, and neutral equilibrium, and it helps in solving the exerciser's push-up problem where forces are exerted on different points (hands and feet) of the man's body.

Rotational Equilibrium

Rotational equilibrium takes the concept of torque equilibrium further by ensuring that an object is not only static in the linear sense but also not undergoing any rotational motion. This means both torque and the linear forces must be balanced.

For an object to be in rotational equilibrium:

• The net force acting in any direction should be zero.
• The net torque around any axis should also be zero.

In the example of the man doing a push-up, we assume there is no angular movement, meaning the man’s body doesn't rotate around his feet or palms. We ensure rotational equilibrium by checking that no unbalanced forces cause the man to tilt forward or backward, helping us calculate how much force his palms and feet should exert to keep everything in balance.

Force Calculation

Calculating forces involves determining the magnitudes of forces acting on an object to achieve equilibrium. It requires understanding the roles of individual forces and their sum must lead to a balanced or stable state.

In the push-up scenario, two primary forces are considered: the force on the palms (\( F_p \)) and the force on the feet (\( F_f \)). The task is to find values for these that, when combined, counteract the man’s weight and keep him steady. This is done by:

• Calculating the total force acting due to gravity (weight).
• Applying torque equilibrium around the man's feet to solve for the force at the palms.
• Using the total force equilibrium to solve for the force at the feet.

Ultimately, understanding force calculations helps us find the exact distribution of forces required to support the body in different positions, as demonstrated with the force exerted by the floor on the man's palms and feet.