Problem 46 - University Physics with Modern Physics

Question

A nail in a pine board stops a 4.9-N hammer head from an initial downward velocity of 3.2 m/s in a distance of 0.45 cm. In addition, the person using the hammer exerts a 15-N downward force on it. Assume that the acceleration of the hammer head is constant while it is in contact with the nail and moving downward. (a) Draw a free-body diagram for the hammer head. Identify the reaction force for each action force in the diagram. (b) Calculate the downward force \(\vec{F}\) exerted by the hammer head on the nail while the hammer head is in contact with the nail and moving downward. (c) Suppose that the nail is in hardwood and the distance the hammer head travels in coming to rest is only 0.12 cm. The downward forces on the hammer head are the same as in part (b). What then is the force \(\vec{F}\) exerted by the hammer head on the nail while the hammer head is in contact with the nail and moving downward?

Step-by-Step Solution

Answer

Part (b): 573.89 N; Part (c): 2123.34 N.

Step 1: Free-Body Diagram - Forces Identification

Identify all the forces acting on the hammer head when it is in contact with the nail. The forces include:1. Weight of the hammer head (\(\vec{W}\ = 4.9\,\text{N}\) downward).2. Force exerted by the person (\(\vec{F}_{\text{person}} = 15\,\text{N}\) downward).3. Normal reaction force (\(\vec{F}_{\text{nail}}\) upward), which is the force exerted by the nail on the hammer head. The free-body diagram will show these forces with the direction of the net force resulting in the change of motion.

Step 2: Apply Newton's Second Law

The net force (\(\vec{F}_{\text{net}}\) exerted on the hammer head can be found using:\[\vec{F}_{\text{net}} = \vec{W} + \vec{F}_{\text{person}} - \vec{F}_{\text{nail}} = m \cdot a\]Here, \(\vec{F}_{\text{nail}}\) is the reaction force we need to find. Since the acceleration (\(a\)) can be determined by the change in velocity and distance, we first find it using kinematic equations.

Step 3: Determine Acceleration Using Kinematics for Part (b)

The hammer's change in motion is governed by the kinematic equation:\[0 = (3.2)^2 + 2 \cdot a \cdot 0.0045\]Solving for \(a\) we find:\[a = -\frac{(3.2)^2}{2 \cdot 0.0045}\ \approx -1137.78 \,\text{m/s}^2\]

Step 4: Calculate Force for Part (b) Using Acceleration

Use the found acceleration to calculate \(\vec{F}_{\text{nail}}\) for part (b):\[\vec{F}_{\text{net}} = m \cdot a = -\vec{F}_{\text{nail}}\]Sum of forces is:\[4.9 + 15 - \vec{F}_{\text{nail}} = 0.5 \cdot -1137.78\]Solving gives:\[\vec{F}_{\text{nail}} \approx 573.89 \,\text{N}\]

Step 5: Determine Acceleration Using Kinematics for Part (c)

Change the distance to 0.0012 m:\[0 = (3.2)^2 + 2 \cdot a \cdot 0.0012\]Solving for \(a\) gives:\[a = -\frac{(3.2)^2}{2 \cdot 0.0012} \approx -4266.67 \,\text{m/s}^2\]

Step 6: Calculate Force for Part (c) Using Acceleration

Use the found acceleration for part (c) to calculate \(\vec{F}_{\text{nail}}\):\[\vec{F}_{\text{net}} = 0.5 \cdot -4266.67 \]Sum of forces is:\[4.9 + 15 - \vec{F}_{\text{nail}} = 0.5 \left(-4266.67\right)\]Solving gives:\[\vec{F}_{\text{nail}} \approx 2123.34 \,\text{N}\]

Key Concepts

Free-Body Diagram

Understanding the free-body diagram is crucial in problem-solving related to forces. In this context, a free-body diagram helps visualize all the forces acting on the hammer head when it is in contact with the nail. These forces include:

• The weight of the hammer head, denoted as \(\vec{W} = 4.9\,\text{N}\), acting downward.
• The force exerted by the person, \(\vec{F}_{\text{person}} = 15\,\text{N}\), also acting downward.
• The normal reaction force from the nail, \(\vec{F}_{\text{nail}}\), which acts upward and opposes the downward motion.

Visualizing these forces with arrows can show the direction of each force. Such diagrams are incredibly useful as they provide a clear representation of the problem. Always remember to identify action-reaction force pairs, as per Newton's Third Law, where every action has an equal and opposite reaction. By doing so, you can precisely determine the resultant force that is acting to alter the motion of the object.

Force Calculation

Force calculation is the essence of applying Newton's Second Law, which tells us that the net force is equal to the mass times the acceleration (F_{\text{net}} = m \cdot a). Calculating forces helps determine how much an object will accelerate when subjected to certain forces.

In our scenario, the hammer head's force exerted on the nail involves summing all forces acting on the hammer:

• The calculated net force is a result of the weight of the hammer, the person's force, and the normal reaction force.
• With the known accelerations for different contact points (e.g., pine board and hardwood), we could calculate the respective net forces involved.
• In cases where acceleration was determined as \(a = -1137.78\,\text{m/s}^2\) and \(a = -4266.67\,\text{m/s}^2\), it provides measurable insights into the scenario.
• Consequently, the nail exerts reaction forces of approximately \(573.89\,\text{N}\) and \(2123.34\,\text{N}\) for the two scenarios.

Calculating such forces can tell us not just about the forces involved, but also how different conditions or materials can drastically influence the outcome.

Kinematic Equations

Kinematic equations serve as a vital tool for analyzing motion in physics, particularly when acceleration is constant. They allow us to connect variables like velocity, acceleration, and displacement, which are all critical to solving force problems. In this case, the equation used is: \[ v_f^2 = v_i^2 + 2a d \] where \(v_f\) is the final velocity, \(v_i\) is the initial velocity, \(a\) is the acceleration, and \(d\) is the distance moved.

By using this equation:

• We set the final velocity (\(v_f\)) to zero because the hammer stops.
• The initial velocity (\(v_i\)) is given as \(3.2\,\text{m/s}\).
• The distance (\(d\)) is converted into meters, for instance, 0.45 cm becomes 0.0045 m.

Solving for the acceleration, we obtained different acceleration values for the scenarios, such as \(-1137.78\,\text{m/s}^2\) for the pine board and \(-4266.67\,\text{m/s}^2\) for the hardwood. Recognizing how kinematic equations connect different physical quantities helps unravel the complexities of motion. This understanding is not only critical for force determination but also for grasping the broader dynamics at play in physical interactions.