Problem 52 - University Physics with Modern Physics

Question

A \(4.9-\mathrm{N}\) hammer head is stopped from an initial downward velucity of 3.2 \(\mathrm{m} / \mathrm{s}\) in a distance of 0.45 \(\mathrm{cm}\) by a nail in a pine board. In addition to its weight, there is a \(15-\mathrm{N}\) downward force on the hammer head applied by the person using the hammer. 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 to each action force in the diagram. (b) Calculate the downward force \(\overrightarrow{\boldsymbol{F}}\) exerted by the hammer head on the nail while the hammer head is in contact with the nail and moving downward. (c) Suppose the nail is in hardwood and the distance the hammer head travels in coming to rest is only 0.12 \(\mathrm{cm} .\) The downward forces on the hammer head are the same as on part (b). What then is the force \(\overrightarrow{\boldsymbol{F}}\) 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

(b) 588.79 N in pine board. (c) 2153.24 N in hardwood.

Step 1: Identify Known Values

The problem gives us known values: - Weight of hammer head: 4.9 N - Initial velocity: 3.2 m/s - Stopping distance in pine board: 0.45 cm = 0.0045 m - Additional force applied: 15 N - Alternate stopping distance in hardwood: 0.12 cm = 0.0012 m

Step 2: Draw Free-Body Diagram

The free-body diagram for the hammer head includes: - Downward forces: weight (4.9 N) and applied force (15 N) - Upward force: normal force from the nail, opposed to the motion of the hammer head.

Step 3: Total Downward Force

The total downward force when the hammer head is in contact with the nail is the sum of the hammer's weight and the additional applied force: \[ F_{down} = 4.9 \, \text{N} + 15 \, \text{N} = 19.9 \, \text{N} \]

Step 4: Calculate Retardation in Pine Board

Using the given, we apply the kinematic equation: \[ v^2 = u^2 + 2as \] Plugging the values, where \(v = 0\), \(u = 3.2 \, \text{m/s}\), and \(s = 0.0045 \, \text{m}\), solve for \(a\). \[ 0 = 3.2^2 + 2a(0.0045) \] \[ a = -\frac{3.2^2}{2 \times 0.0045} = -1137.78 \, \text{m/s}^2 \]

Step 5: Calculate Force in Pine Board

Apply Newton's second law: \[ F_{net} = ma \] The net force \(F_{net} = F_{down} - F\). Thus, \[ F = F_{down} - ma = 19.9 - \frac{4.9}{9.8} \times 1137.78 \] Calculating mass, \(m = \frac{4.9}{9.8} \approx 0.5 \, \text{kg}\), results in: \[ F \approx 19.9 + 568.89 = 588.79 \, \text{N} \]

Step 6: Calculate Retardation in Hardwood

Using the same initial velocity equation in hardwood: \[ 0 = 3.2^2 + 2a(0.0012) \] \[ a = -\frac{3.2^2}{2 \times 0.0012} = -4266.67 \, \text{m/s}^2 \]

Step 7: Calculate Force in Hardwood

Apply Newton's second law again for hardwood: \[ F_{net} = ma \] Similarly, \[ F = F_{down} - ma = 19.9 - 0.5 \times 4266.67 \] \[ F \approx 19.9 + 2133.335 = 2153.235 \, \text{N} \]

Key Concepts

Free-body Diagram

When you picture a hammer head moving towards a nail, it's important to identify all the forces acting on it. This is where the free-body diagram comes in handy. A free-body diagram is a visual tool that helps you analyze the forces acting on an object. For the hammer head in this exercise, there are two key forces at play:

• The weight of the hammer head itself, which is given as 4.9 N and acts downward.
• An additional force of 15 N applied by the person using the hammer, which also acts downward.

Together, these forces create a total downward force. There is also an upward force:

• The normal force from the nail, opposing the action of the hammer head exerted upward.

In the context of the free-body diagram, each force can be thought of as an action with a corresponding reaction force, according to Newton's third law. The key takeaway from drawing a free-body diagram is understanding how forces interact and balance each other.

Kinematic Equations

Kinematic equations are vital in motion analysis and can simplify complex problems involving acceleration, velocity, and displacement. In this scenario, the hammer head is brought to rest over a short distance as it contacts the nail. These equations allow us to relate initial velocity, final velocity, acceleration, and distance.One key equation is:\[ v^2 = u^2 + 2as \]Where:

• \(v\) is the final velocity (0 m/s, since the hammer head stops).
• \(u\) is the initial velocity (3.2 m/s).
• \(a\) is the acceleration (or deceleration, as it stops).
• \(s\) is the stopping distance.

This equation helps you calculate the deceleration experienced by the hammer head in different materials (pine and hardwood). Deceleration is negative acceleration, showing how quickly something slows down. By knowing this, you can calculate the force exerted by the hammer head on the nail.

Newton's Second Law

Newton's second law of motion connects force, mass, and acceleration through the equation:\[ F = ma \]In simple terms, this law establishes that the force acting on an object is equal to the object's mass multiplied by its acceleration. In this exercise, you're looking at the hammer head in terms of the forces involved while it interacts with the nail.Firstly, you determine the hammer head's mass using its weight (4.9 N), as weight is the force due to gravity, given by \(mg\), where \(g = 9.8\, \text{m/s}^2\). Solving for mass provides:\[ m = \frac{4.9}{9.8} \approx 0.5 \text{ kg} \]Using the calculated mass and the deceleration obtained from kinematic equations, you can assess the net force required to stop the hammer.Newton's second law helps determine the effective force exerted by the hammer head on the nail during the deceleration process, different in pine and hardwood due to the varied stopping distances. The understanding of how force, mass, and acceleration interplay is crucial in predicting the forces encountered in real-world situations.