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Chapter 14: Problem 22

I A nail driven into a board increases in temperature. If \(60 \%\) of the kinetic energy delivered by a \(1.80 \mathrm{~kg}\) hammer with a speed of \(7.80 \mathrm{~m} / \mathrm{s}\) is transformed into heat that flows into the nail and does not flow out, what is the increase in temperature of an \(8.00 \mathrm{~g}\) aluminum nail after it is struck 10 times?

Short Answer

Expert verified
The nail's temperature increases by approximately 45.63 掳C.

Step by step solution

01

Calculate the Kinetic Energy of the Hammer

First, find the kinetic energy of the hammer using the formula: \[ KE = \frac{1}{2} m v^2 \]where \( m \) is the mass of the hammer (1.80 kg) and \( v \) is the velocity (7.80 m/s).Substitute the given values:\[ KE = \frac{1}{2} \times 1.80 \times (7.80)^2 \]\[ KE = \frac{1}{2} \times 1.80 \times 60.84 \]\[ KE = 54.756 \, \text{Joules} \]
02

Determine the Heat Transferred to the Nail

Given that 60% of the kinetic energy is transferred to the nail as heat, calculate that amount:\[ Q = 0.60 \times KE \]\[ Q = 0.60 \times 54.756 \]\[ Q = 32.8536 \, \text{Joules} \]This is the energy transferred to the nail during one strike.
03

Calculate Total Energy from Multiple Strikes

The nail is struck 10 times, so calculate the total energy transferred after these strikes:\[ Q_{\text{total}} = 10 \times Q \]\[ Q_{\text{total}} = 10 \times 32.8536 \]\[ Q_{\text{total}} = 328.536 \, \text{Joules} \]
04

Use the Heat Capacity Formula to Find Temperature Change

To find the temperature increase, use the formula:\[ \Delta T = \frac{Q_{\text{total}}}{m c} \]where \( m \) is the mass of the nail in kg (8.00 g = 0.008 kg) and \( c \) is the specific heat capacity of aluminum (\( 900 \, \text{J/kg} \, ^\circ C \)).Substitute the values:\[ \Delta T = \frac{328.536}{0.008 \times 900} \]\[ \Delta T = \frac{328.536}{7.2} \]\[ \Delta T = 45.63 \, ^\circ C \]
05

Conclusion

The temperature of the aluminum nail increases by approximately \( 45.63 \, ^\circ C \) after being struck 10 times by the hammer.

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Key Concepts

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

Kinetic Energy
Kinetic energy is the energy that an object possesses due to its motion. It depends on two main factors: the mass of the object and its velocity.
  • The formula to calculate kinetic energy is \( KE = \frac{1}{2} m v^2 \).
  • Where \( m \) is mass, and \( v \) is velocity.
In the original exercise, a hammer is struck with a mass of 1.80 kg and a velocity of 7.80 m/s. By plugging these values into the formula, we determine that the hammer鈥檚 kinetic energy is approximately 54.76 Joules. Kinetic energy plays a crucial role in many physical processes, and its conservation is pivotal because when energy is not conserved, it often finds a way to be transformed into other forms, such as heat or potential energy.
Heat Transfer
Heat transfer is the movement of thermal energy from one object to another. It occurs only when there is a temperature difference between the two objects and can happen through conduction, convection, or radiation.
  • In this scenario, heat is transferred from the hammer to the nail.
  • This is an example of conduction, where thermal energy is directly transferred between surfaces in contact.
  • The heat transferred from the kinetic energy of the hammer to the nail increases the nail's temperature.
In the exercise, we learned that 60% of the hammer's kinetic energy is transferred as heat into the nail, meaning approximately 32.85 Joules per strike. Repeatedly striking the nail (10 times) results in a total heat transfer of 328.54 Joules.
Specific Heat Capacity
Specific heat capacity is a property that tells us how much heat energy is needed to raise the temperature of a unit mass of a substance by one degree Celsius.
  • The formula to calculate the change in temperature is \( \Delta T = \frac{Q}{m c} \), where:
  • \( Q \) is the total heat energy transferred,
  • \( m \) is the mass of the substance, and
  • \( c \) is the specific heat capacity.
For aluminum, which our nail is made of, the specific heat capacity is 900 J/kg 掳C. Given the heat transferred (328.54 Joules), mass of the nail (0.008 kg), and aluminum's specific heat capacity, the temperature of our nail increases by approximately 45.63 掳C. This property helps us understand why different substances heat up at different rates when the same amount of energy is applied.

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Most popular questions from this chapter

A copper pot with a mass of \(0.500 \mathrm{~kg}\) contains \(0.170 \mathrm{~kg}\) of water, and both are at a temperature of \(20.0^{\circ} \mathrm{C}\). A \(0.250 \mathrm{~kg}\) block of iron at \(85.0^{\circ} \mathrm{C}\) is dropped into the pot. Find the final temperature of the system, assuming no heat loss to the surroundings.

| From a height of \(35.0 \mathrm{~m}\), a \(1.25 \mathrm{~kg}\) bird dives (from rest) into a small fish tank containing \(50.0 \mathrm{~kg}\) of water. What is the maximum rise in temperature of the water if the bird gives it all of its mechanical energy?

A box-shaped coal-burning stove has exhausted most of its fuel, and its surface temperature has fallen to \(27^{\circ} \mathrm{C}\). After more coal is added, the surface temperature eventually rises to \(327^{\circ} \mathrm{C}\). By what factor does the stove's radiation heat transfer to the surroundings increase after the coal is added?

for this temperature? (b) Elevated body temperature. During very vigorous exercise, the body's temperature can go as high as \(40^{\circ} \mathrm{C}\). What would Kelvin and Fahrenheit thermometers read for this temperature? (c) Temperature difference in the body. The surface temperature of the body is normally about \(7 \mathrm{C}^{\circ}\) lower than the internal temperature. Express this temperature difference in kelvins and in Fahrenheit degrees. (d) Blood storage. Blood stored at \(4.0^{\circ} \mathrm{C}\) lasts safely for about 3 weeks, whereas blood stored at \(-160^{\circ} \mathrm{C}\) lasts for 5 years. Express both temperatures on the Fahrenheit and Kelvin scales. (e) Heat stroke. If the body's temperature is above \(105^{\circ} \mathrm{F}\) for a prolonged period, heat stroke can result. Express this temperature on the Celsius and Kelvin scales.

A carpenter builds an exterior house wall with a layer of wood \(3.0 \mathrm{~cm}\) thick on the outside and a layer of Styrofoam insulation \(2.2 \mathrm{~cm}\) thick on the inside wall surface. The wood has a thermal conductivity of \(0.080 \mathrm{~W} /(\mathrm{m} \cdot \mathrm{K}),\) and the Styrofoam has a thermal conductivity of \(0.010 \mathrm{~W} /(\mathrm{m} \cdot \mathrm{K})\). The interior surface temperature is \(19.0^{\circ} \mathrm{C}\). and the exterior surface temperature is \(-10.0^{\circ} \mathrm{C}\). (a) What is the temperature at the plane where the wood meets the Styrofoam? (b) What is the rate of heat flow per square meter through this wall?
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