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 first calculate the center of gravity (CG) location and then use the concept of torque equilibrium.
The first step is to determine the CG location. We are given that the man's hands are 36 cm apart and his CG is located 85% of the distance from his right hand towards his left.
Let's calculate the total distance between his hands:
Total distance = 36 cm
Now, we need to find the location of the CG:
Distance from right hand (85%) = 0.85 * 36 cm = 30.6 cm
Since the man's CG is located 85% of the distance from his right hand toward his left, the CG is 30.6 cm from his right hand.
Next, we need to calculate the force on each hand due to the ground. We will use the concept of torque equilibrium, which states that the sum of the torques acting on a system is zero.
In this case, the only torque acting on the system is the torque due to the weight of the man, which can be represented by the force due to gravity acting at his CG.
Now, let's assume that each hand exerts a force F on the ground. The torque exerted by the man's weight acting at his CG will be equal to the torque exerted by the forces on his hands.
The torque exerted by the man's weight (M) = force (F) * distance (d)
The torque exerted by the man's weight can be calculated using the formula:
Torque = weight (W) * distance (d)
Since weight (W) = mass (m) * gravitational acceleration (g), we have:
Torque = m * g * d
The torque exerted by the man's weight (M) can be written as:
M = m * g * d
Now, let's set up the equation for torque equilibrium:
Torque exerted by the forces on the hands = Torque exerted by the man's weight
Force on right hand * distance from right hand = Force on left hand * distance from left hand
Force on right hand * (36 cm - distance from right hand) = Force on left hand * distance from left hand
Let's substitute the values we found:
Force on right hand * (36 cm - 30.6 cm) = Force on left hand * 30.6 cm
Force on right hand * 5.4 cm = Force on left hand * 30.6 cm
Now, we need to find the ratio of the forces by dividing one equation by the other:
Force on right hand / Force on left hand = (30.6 cm) / (5.4 cm)
Force on right hand / Force on left hand = 5.67
Therefore, the force on the right hand is approximately 5.67 times the force on the left hand.