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Chapter 11: Problem 52

A claw hammer is used to pull a nail out of a board (see Fig. \(\mathrm{P} 11.45\) ). The nail is at an angle of \(60^{\circ}\) to the board, and a force \(\overrightarrow{\boldsymbol{F}}_{1}\) of magnitude \(400 \mathrm{~N}\) applied to the nail is required to pull it from the board. The hammer head contacts the board at point \(A,\) which is \(0.080 \mathrm{~m}\) from where the nail enters the board. A horizontal force \(\vec{F}_{2}\) is applied to the hammer handle at a distance of \(0.300 \mathrm{~m}\) above the board. What magnitude of force \(\overrightarrow{\boldsymbol{F}}_{2}\) is required to apply the required \(400 \mathrm{~N}\) force \(\left(F_{1}\right)\) to the nail? (Ignore the weight of the hammer.)

Short Answer

Expert verified
The magnitude of the force \(F_{2}\) required to apply the required 400N force to the nail is approximately 106.67N.

Step by step solution

01

Identify the forces and distances

First a few forces and distances should be identified: The force that needs to be exerted on the nail, \(F_{1} = 400N\), the distance between the nail and the point where the hammer makes contact with the board, \(d_{1} = 0.080m\), and the distance from the point where the force is applied on the hammer to the point of contact on the board, \(d_{2} = 0.300m\).
02

Apply the principle of moments

The principle of moments states that for a lever in equilibrium, the total clockwise moment about any point is equal to the total anticlockwise moment about the same point. In this case, the anticlockwise moment is \(F_{1} \cdot d_{1}\) and the clockwise moment is \(F_{2} \cdot d_{2}\) for the force \(F_{2}\) we are trying to find. These two moments are equal, so \(F_{1} \cdot d_{1} = F_{2} \cdot d_{2}\).
03

Solve for \(F_{2}\)

Solving the equation \(F_{1} \cdot d_{1} = F_{2} \cdot d_{2}\) for \(F_{2}\) yields \(F_{2} = F_{1} \cdot \frac{d_{1}}{d_{2}}\). Plugging in the given values leads to \(F_{2} = 400N \cdot \frac{0.080m}{0.300m} = 106.67N\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Principle of Moments
The principle of moments is a fundamental concept in physics that helps us understand how forces cause objects to rotate around a pivot or fulcrum. In simple terms, it tells us that for an object to be in equilibrium, the total moment (or torque) exerted by forces trying to rotate the object in one direction must balance the total moment exerted by forces trying to rotate it in the opposite direction.

This principle can be expressed with the formula:
  • Clockwise moment = Anticlockwise moment
In the context of lever systems, the moment is calculated by multiplying the force applied by the perpendicular distance from the pivot to the line of action of the force. If you hold a ruler on its edge and push down on one end, you can feel the turning effect at the pivot. This is the moment created by your force.

In our problem, the moment caused by the force pulling the nail (\(F_1dot \) \(d_1\)) is trying to turn the hammer counterclockwise. While the force you're applying, \(F_2\), multiplied by its distance from the pivot (\(d_2\)) tries to rotate it clockwise. Using the principle of moments, these two moments must be equal to keep the system from spinning out of control.
Equilibrium
The concept of equilibrium in mechanics refers to the state where all forces and moments acting on a system cancel each other out. When a system is in equilibrium:
  • The sum of all forces equals zero (no net force), ensuring there is no linear movement.
  • The sum of all moments equals zero (no net moment), ensuring there is no rotation.
In our exercise, the claw hammer and nail system must reach a state of equilibrium to function correctly. Here, equilibrium ensures that the force you apply (\(F_2\)) is perfectly balanced with the force needed to pull the nail (\(F_1\)).

To maintain equilibrium, we must choose the proper force to apply to the hammer handle. By setting the clockwise and anticlockwise moments equal to each other, the system achieves rotational equilibrium. This balance prevents the hammer from twisting too much, allowing it to maximally transfer the exerted force to the nail.
Lever Systems
Lever systems are simple machines that allow us to gain mechanical advantage in applying force. A lever consists of a beam or rod pivoted at a fulcrum. By applying a force at a distance from the pivot, you can move an object with less effort than would otherwise be required.

The mechanics of lever systems are dictated by three main components:
  • The effort or force you apply, \(F_2\) in this instance.
  • The load or resistance force, such as \(F_1\) in our example.
  • The fulcrum or pivot point.
In the exercise, the claw hammer acts as a lever. When you apply force on the handle, the hammer pivots at the point where it contacts the board. The hammer amplifies your input force, allowing the nail to be pulled out with less physical effort. By positioning the fulcrum closer to the load (the nail), the lever allows you to apply lesser force (\(F_2\)) over a longer distance (\(d_2\)). This is the principle behind using a lever: using distance to improve force efficacy and efficiency. This way, a small input can cause a significant output."

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Most popular questions from this chapter

A uniform, 7.5-m-long beam weighing \(6490 \mathrm{~N}\) is hinged to a wall and supported by a thin cable attached \(1.5 \mathrm{~m}\) from the free end of the beam. The cable runs between the beam and the wall and makes a \(40^{\circ}\) angle with the beam. What is the tension in the cable when the beam is at an angle of \(30^{\circ}\) above the horizontal?

A metal rod that is \(4.00 \mathrm{~m}\) long and \(0.50 \mathrm{~cm}^{2}\) in crosssectional area is found to stretch \(0.20 \mathrm{~cm}\) under a tension of \(5000 \mathrm{~N}\). What is Young's modulus for this metal?

A \(72.0 \mathrm{~kg}\) weightlifter doing arm raises holds a \(7.50 \mathrm{~kg}\) weight. Her arm pivots around the elbow joint, starting \(40.0^{\circ}\) below the horizontal (Fig. \(\mathbf{P 1 1 . 5 8}\) ). Biometric measurements have shown that, together, the forearms and the hands account for \(6.00 \%\) of a person's weight. since the upper arm is held vertically, the biceps muscle always acts vertically and is attached to the bones of the forearm \(5.50 \mathrm{~cm}\) from the elbow joint. The center of mass of this person's forearm-hand combination is \(16.0 \mathrm{~cm}\) from the elbow joint, along the bones of the forearm, and she holds the weight \(38.0 \mathrm{~cm}\) from her elbow joint. (a) Draw a freebody diagram of the forearm. (b) What force does the biceps muscle exert on the forearm? (c) Find the magnitude and direction of the force that the elbow joint exerts on the forearm. (d) As the weightlifter raises her arm toward a horizontal position, will the force in the biceps muscle increase, decrease, or stay the same? Why?

Two people carry a heavy electric motor by placing it on a light board \(2.00 \mathrm{~m}\) long. One person lifts at one end with a force of \(400 \mathrm{~N}\), and the other lifts the opposite end with a force of \(600 \mathrm{~N}\). (a) What is the weight of the motor, and where along the board is its center of gravity located? (b) Suppose the board is not light but weighs \(200 \mathrm{~N},\) with its center of gravity at its center, and the two people exert the same forces as before. What is the weight of the motor in this case, and where is its center of gravity located?

You are a construction engineer working on the interior design of a retail store in a mall. A 2.00 -m-long uniform bar of mass \(8.50 \mathrm{~kg}\) is to be attached at one end to a wall, by means of a hinge that allows the bar to rotate freely with very little friction. The bar will be held in a horizontal position by a light cable from a point on the bar (a distance \(x\) from the hinge) to a point on the wall above the hinge. The cable makes an angle \(\theta\) with the bar. The architect has proposed four possible ways to connect the cable and asked you to assess them: $$ \begin{array}{lllll} \text { Alternative } & \text { A } & \text { B } & \text { C } & \text { D } \\\ \hline x(\mathrm{~m}) & 2.00 & 1.50 & 0.75 & 0.50 \\ \theta(\text { degrees }) & 30 & 60 & 37 & 75 \end{array} $$ (a) There is concern about the strength of the cable that will be required. Which set of \(x\) and \(\theta\) values in the table produces the smallest tension in the cable? The greatest? (b) There is concern about the breaking strength of the sheetrock wall where the hinge will be attached. Which set of \(x\) and \(\theta\) values produces the smallest horizontal component of the force the bar exerts on the hinge? The largest? (c) There is also concern about the required strength of the hinge and the strength of its attachment to the wall. Which set of \(x\) and \(\theta\) values produces the smallest magnitude of the vertical component of the force the bar exerts on the hinge? The largest? (Hint: Does the direction of the vertical component of the force the hinge exerts on the bar depend on where along the bar the cable is attached?) (d) Is one of the alternatives given in the table preferable? Should any of the alternatives be avoided? Discuss.
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