Figure 12–53 shows a pair of forceps used to hold a thin plastic rod firmly. If the thumb and finger each squeeze with a force FT = FF = 11.0 N, what force do the forceps jaws exert on the plastic rod?
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15. Rotational Equilibrium
Torque & Equilibrium
Problem 60
Textbook Question
A woman holds a 1.8-m-long uniform 10.0-kg pole as shown in Fig. 12–84. (a) Determine the forces she must exert with each hand (magnitude and direction). To what position should she move her left hand so that neither hand has to exert a force greater than (b) 150 N? (c) 85 N?
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Step 1: Identify the forces acting on the pole. The pole is uniform, so its weight (W = m * g) acts at its center of gravity, which is at the midpoint of the pole. The forces exerted by the woman's hands (F_left and F_right) act at the positions of her hands. Assume the left hand is at a distance x from the left end of the pole, and the right hand is at the far end of the pole (1.8 m from the left end).
Step 2: Apply the conditions for static equilibrium. For the pole to remain stationary, the sum of all vertical forces must equal zero, and the sum of all torques about any point must also equal zero. Mathematically, these conditions are: ΣF_y = 0 (vertical force balance) and Στ = 0 (torque balance).
Step 3: Write the equations for vertical force balance. The total upward forces exerted by the hands must equal the downward gravitational force of the pole: F_left + F_right = W, where W = 10.0 kg * 9.8 m/s².
Step 4: Write the torque balance equation. Choose the left hand as the pivot point. The torque due to the pole's weight is W * (1.8 m / 2), and the torque due to the right hand is F_right * (1.8 m - x). Since the system is in equilibrium, the sum of torques is zero: W * (1.8 m / 2) = F_right * (1.8 m - x).
Step 5: Solve the equations. Use the two equations (ΣF_y = 0 and Στ = 0) to solve for F_left and F_right in terms of x. For parts (b) and (c), adjust the position of the left hand (x) such that neither F_left nor F_right exceeds the given limits (150 N for part b and 85 N for part c). This involves substituting the limits into the equations and solving for x.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Torque
Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point (lever arm). In this scenario, the woman must balance the torques acting on the pole to determine the forces exerted by her hands. Understanding how to calculate and balance torques is essential for solving the problem.
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Equilibrium
Equilibrium refers to a state where the sum of forces and the sum of torques acting on an object are both zero. For the woman holding the pole, this means that the upward forces exerted by her hands must equal the downward force due to the pole's weight, and the torques around any pivot point must also balance. This concept is crucial for determining the forces exerted by her hands.
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Force Distribution
Force distribution involves understanding how forces are shared among different points of contact. In this case, the woman must adjust the position of her hands to ensure that the forces exerted do not exceed specified limits. Analyzing how the weight of the pole and the positions of her hands affect the forces is key to solving parts (b) and (c) of the question.
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