Suppose you hit a steel nail with a 0.500 -kg hammer,...

Suppose you hit a steel nail with a 0.500 -kg hammer, initially moving at 15.0 $\mathrm{m} / \mathrm{s}$ and brought to rest in 2.80 $\mathrm{mm}$ . (a) What average force is exerted on the nail? (b) How much is the nail compressed if it is 2.50 $\mathrm{mm}$ in diameter and 6.00 -cm long? (c) What pressure is created on the $1.00-\mathrm{mm}-$ diameter tip of the nail?

Key Concepts

Impulse-Momentum Theorem

This theorem explains that the change in momentum of an object is equal to the impulse applied to it, which is the product of the force and the time interval over which it acts. It is particularly useful in analyzing collisions where forces act over very short periods.

Work-Energy Principle

The work-energy principle states that the total work done on an object is equal to its change in kinetic energy. When a force is applied over a distance, this principle helps relate the force to the energy transformation taking place during the process, such as deceleration.

Elastic Deformation (Hooke’s Law)

Elastic deformation refers to the temporary change in shape of a material under stress, where the deformation is directly proportional to the applied force up to the elastic limit. Hooke’s Law characterizes this behavior by relating force to displacement, which is essential in analyzing compression in structures.

Pressure Calculation

Pressure is defined as the force applied per unit area. It provides a measure of the intensity of the force distributed over a surface, which is crucial when assessing the stress or potential damage on materials due to concentrated forces.

Transcript

00:01 We're told that we hit a steel nail with a 0 .5 kilogram hammer that's initially moving at 15 meters per second and brought the rest in 2 .8 millimeters.

00:11 So first we're asked what the average force is exerted on the nail.

00:16 So i've drawn a little diagram here.

00:18 We have our hammer coming down at 15 meters per second.

00:22 It drives the nail into the wood or whatever it is at a distance 2 .8 millimeters.

00:31 And at this point the hammer is at rest.

00:34 So we want to figure out what the average force is.

00:37 To do that, we need to figure out what the average acceleration is, or deceleration is.

00:43 So we have our values here, and our average acceleration is the change in velocity divided by the change in time.

00:52 Well, we can rewrite that as the change in velocity divided by the change in displacement, multiplied by the change in displacement, divided by the change in time.

01:03 So we have the change in velocity.

01:05 In the end, we get the change in velocity squared divided by the change in displacement, the distance.

01:18 So we can then plug our values in, and we get that a rather large acceleration of 80 ,400 meters per second squared, which gives us if the hammer weighs half a kilogram that gives us a force average force of 40 .2 kilontons.

01:43 Now we're asked how much is the nail compressed if it is two and a half millimeters a diameter and six centimeters long...

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