The 66 kg man's hands in Figure 9-56 are 36 cm apart. His CG is located 85% of the distance from his right hand toward his left. Find the force on each hand due to the ground.
To find the force on each hand due to the ground, we need to consider the concept of torque.
Torque is defined as the tendency of a force to rotate an object about an axis. The torque equation is given by:
τ = F * r * sin(θ)
where τ is the torque, F is the force, r is the distance from the axis of rotation, and θ is the angle between the force vector and the lever arm.
In this case, we can consider the axis of rotation to be the man's left hand. We want to find the force on each hand due to the ground, so we need to find the torque on each hand.
Let's start by finding the torque on the right hand. Since the center of gravity (CG) is located 85% of the distance from his right hand towards his left, we can calculate the distance from his right hand to his CG as follows:
Distance from right hand to CG = 36 cm * (1 - 0.85) = 36 cm * 0.15 = 5.4 cm = 0.054 m
Now, we need to find the torque on the right hand. Since the force due to the ground acts directly below the CG, the angle between the force and the lever arm is 90 degrees. Therefore, the torque on the right hand is:
τ_right = F_right * r_right * sin(90 degrees) = F_right * r_right
Similarly, for the left hand, the distance from the CG to the left hand is 36 cm * 0.85 = 30.6 cm = 0.306 m. The torque on the left hand is:
τ_left = F_left * r_left * sin(90 degrees) = F_left * r_left
Here's the key step: Since the man is in equilibrium (not rotating), the sum of the torques on both hands must be zero. So, we can write the equation as:
τ_right + τ_left = 0
F_right * r_right + F_left * r_left = 0
Since the r_left is negative (opposite direction), we can write it as:
F_right * r_right - F_left * |r_left| = 0
Now, let's substitute the values we have:
F_right * 0.054 m - F_left * 0.306 m = 0
Finally, we need one more piece of information to solve this equation: the total force on the man due to the ground, or the weight of the man.
Weight = mass * gravity
Given that the weight of the man is the force due to the ground, we can write it as:
Weight = 66 kg * 9.8 m/s^2 = 646.8 N
Now, we can substitute this value into the equation and solve for the force on each hand:
F_right * 0.054 m - F_left * 0.306 m = 646.8 N
This equation can be solved to determine the force on each hand due to the ground.