Question

A hammerhead of mass \(m=2.00 \mathrm{~kg}\) is allowed to fall onto a nail from a height \(h=0.400 \mathrm{~m} .\) Calculate the maximum amount of work it could do on the nail.

Short answer

Answer: The maximum amount of work the hammerhead could do on the nail is 7.848 Joules.

Step-by-step solution

Identify the given information

In this problem, the mass of the hammerhead (m) is 2.00 kg and the height (h) it falls from is 0.400 m.

Calculate the gravitational potential energy

To calculate the gravitational potential energy (PE), we can use the following equation: PE = m * g * h where m is the mass of the hammerhead, g is the acceleration due to gravity (approximately 9.81 \(\mathrm{m/s^2}\)), and h is the height.

Plug in the given values

Now, we can plug in the given values into the equation: PE = 2.00 kg * 9.81 \(\mathrm{m/s^2}\) * 0.400 m

Calculate the gravitational potential energy

With the values in place, we can perform the calculation: PE = 7.848 kg\(\mathrm{m^2/s^2}\)

Express the gravitational potential energy as work

Since gravitational potential energy and work have the same units, we can directly express the potential energy as the maximum work that can be done by the hammerhead on the nail: Maximum work = 7.848 J Therefore, the maximum amount of work the hammerhead could do on the nail is 7.848 Joules.

Key concepts

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. Think of it as stored energy that can be converted to other forms of energy, such as kinetic energy. When you hold an object at a certain height, you're storing energy in it. This energy depends on three main factors:

• The mass of the object
• Its height above the ground
• The gravitational pull acting on it (which on Earth is approximately 9.81 m/s²)

To calculate the gravitational potential energy (PE), use the formula:\[PE = m \times g \times h\]Here, \(m\) is the mass in kilograms, \(g\) is the gravitational acceleration, and \(h\) is the height in meters. In our exercise, the hammerhead has mass \(m = 2.00 \, \text{kg}\), falls from height \(h = 0.400 \, \text{m}\), and the gravitational pull is roughly \(9.81 \, \text{m/s}^2\). Plug these values into the formula to get the gravitational potential energy:\[PE = 2.00 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 0.400 \, \text{m} = 7.848 \, \text{J}\]

Work Calculation

Work is an essential concept in physics representing the energy transferred by a force. When force acts upon an object causing it to move, work is done. The hammerhead in our scenario exemplifies this as it descends under gravity, doing work on the nail. The calculation for the maximum work involves understanding that gravitational potential energy can be entirely converted into work. Given that energy can not be created nor destroyed, the potential energy lost by the hammerhead as it falls is equal to the work done on the nail. When you calculate work, consider:

• Force applied
• Distance over which the force is applied
• Direction, as work is maximal when force direction matches the motion

In our example, since the hammerhead directly converts its potential energy to work, the maximum work done on the nail is equivalent to the calculated gravitational potential energy of 7.848 Joules.

Energy Conversion

Energy conversion is the changing of energy from one form to another. In physics problems, this is fundamental for understanding system dynamics. For the hammerhead, when it falls, this energy transformation is visible. Initially, it has gravitational potential energy due to its height, which upon the fall is transformed into kinetic energy. This process gives rise to some essential insights:

• Energy conservation: energy is transferred, not lost
• Potential energy decreases as kinetic energy increases during the hammerhead's fall
• The total energy remains the same through the fall

Understanding this conversion helps explain how the potential energy results in practical works such as driving a nail into a surface. The complete transfer emphasizes the role of gravitational potential energy as an energy source that can do work.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. When the hammerhead falls, it accelerates due to gravity, thus gaining speed and kinetic energy.The conversion from potential to kinetic energy is a key process.We calculate kinetic energy (KE) using the formula:\[KE = \frac{1}{2} m v^2\]Here, \(m\) represents mass and \(v\) its velocity. Before the impact, all the gravitational potential energy converts to kinetic energy, allowing maximum work to be done on the nail. This fully dynamic energy is what ultimately drives the nail into the surface.Overall, kinetic energy shows us the movement's energy, demonstrating how energy stored in a static position transforms to fulfill a functional purpose in physics problems.