/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 A metal block of houl capacity \... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 16

A metal block of houl capacity \(80 \mathrm{~J} /{ }^{\circ} \mathrm{C}\) being heated cloctrically is placod in a room at \(20^{\circ} \mathrm{C}\). The heater is switched off when temperature of the block reaches \(30^{\circ} \mathrm{C} .\) The temperature of block rises at rate \(2^{\circ} \mathrm{C} / \mathrm{sJust}\) alter heater was switched ON and falls at the rate \(0.2^{\circ} \mathrm{C} / \mathrm{s}\) when switch OFF. (Assume Newton's law of cooling holds). Find power of heater (a) \(120 \mathrm{~W}\) (b) \(140 \mathrm{~W}\) (c) \(160 \mathrm{~W}\) (d) \(200 \mathrm{~W}\)

Short Answer

Expert verified
The power of the heater is 160 W, option (c).

Step by step solution

01

Understanding the Problem

We are given a metal block with a heat capacity of 80 J/掳C. It starts at 20掳C, is heated until it reaches 30掳C, and then cools at a known rate when the heater is turned off. We must determine the power of the electric heater.
02

Apply Heat Transfer Formula

The rate at which temperature increases when the heater is on is 2掳C/s. Since the heat capacity of the block is 80 J/掳C, the power provided by the heater can be calculated using the formula: \[ P = C \cdot \frac{dT}{dt} \]where \( C = 80 \text{ J/掳C} \) and \( \frac{dT}{dt} = 2 \text{掳C/s} \).
03

Calculate Power of the Heater

Substitute the known values into the formula:\[ P = 80 \text{ J/掳C} \times 2 \text{掳C/s} = 160 \text{ W} \]Therefore, the power of the heater is 160 W.

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Key Concepts

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

Heat Capacity
Heat capacity is an essential concept when dealing with temperature changes in physical objects. It refers to the amount of energy required to change the temperature of an object by one degree Celsius. This concept helps us understand how quickly or slowly a material heats up or cools down.
  • Heat capacity is measured in joules per degree Celsius (\( \text{J/掳C} \)).
  • This property is unique for different substances depending on their masses and material compositions.
  • For instance, a metal block with a heat capacity of 80 J/掳C means it needs 80 joules of energy to change its temperature by 1掳C.
When calculating heat interactions, heat capacity allows us to predict how much energy is necessary to heat or cool an object by a certain amount. It is crucial for efficient energy management and optimizing heating processes.
Newton's Law of Cooling
Newton's Law of Cooling describes the rate at which an exposed body changes temperature through radiation, depending on the difference between the body鈥檚 temperature and the ambient environment. This physical law is crucial for understanding the cooling process when an external heat source is removed.
  • The rate of cooling is proportional to the difference between the object's temperature and its surroundings.
  • The formula is often represented as: \[ \frac{dT}{dt} = -k(T - T_a) \]where \( dT/dt \) expresses the rate of temperature change, \( T \) is the object's temperature, \( T_a \) is the ambient temperature, and \( k \) is a constant.
  • In our exercise, after the heater is switched off at 30掳C, the block cools down, demonstrating this law with a cooling rate of 0.2掳C/s in a room at 20掳C.
Understanding this law helps in predicting and modeling the cooling behavior of objects in different environments.
Power of Electric Heater
The power of an electric heater can be calculated by understanding how heat energy is transferred into a system. Power is the rate at which energy is used or produced, crucial for determining how effective a heater is at increasing temperature.
  • The power generated by a heater relates directly to the heat capacity and the rate of temperature change.
  • The formula used is \[ P = C \times \frac{dT}{dt} \]where \( P \) is the power in watts, \( C \) is the heat capacity in J/掳C, and \( \frac{dT}{dt} \) is the rate of temperature increase in 掳C/s.
  • Using our example, a heater that causes a 2掳C/s increase with a block of 80 J/掳C has a power output of 160 W.
Having a grasp of this concept aids in solving problems related to heating systems and allows for more efficient design and operation of heating appliances.

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Most popular questions from this chapter

A black body is at temperature of \(2880 \mathrm{~K}\). The energy of radiation emitted by this object between wavelength \(4990 \AA\) and \(5000 \AA\) is \(U_{1}\), between \(9900 \AA\) and \(10000 \AA\) is \(U_{2}\) and between \(14990 \AA\) and \(15000 \AA\) is \(U_{3}\), The Wein's constant is \(b=2.88 \times 10^{-3} \mathrm{~m} \mathrm{~K}\) then (a) \(U_{2}>U_{1}\) (b) \(U_{2}>U_{3}\) (c) \(U_{1}=U_{3}

Ice at \(0^{\circ} \mathrm{C}\) is added to \(200 \mathrm{~g}\) of water initially at \(70^{\circ} \mathrm{C}\) in a vacuum flask. When \(50 \mathrm{~g}\) of icc has been added and has all melted the tempcrature of the flask and contents is \(40{ }^{\circ} \mathrm{C}\). When a further \(80 \mathrm{~g}\) of ice has been added and has all melted the temperature of the whole becomes \(10^{\circ} \mathrm{C}\). Find the latent heat of fusion of ice.

A highly conducting hollow cylinder of radius \(a\) and length \(L\) is surrounded by a co-axial layer of a material having thertnal conductivity \(K\) and negligible heat copacity, Temperature of surrounding space (out side the layer) is \(\theta_{2}\), which is higher than temperature of the cylinder. If heat capacity per unit volume of cylinder material is \(s\) and outer radius of tho layer is \(b\), calculate the thermal resistance of the cylindrical shell and also calculate the variation o " lemperaturc inside it with respect to the radial distance from the axis. Assumc cnd faces to be thermally insulaled.

A calorimeter contains \(250 \mathrm{gm}\) of water and \(15 \mathrm{gm}\) of ice at 0 "C. Now steam at 100 " \(\mathrm{C}\) is passed into it continuously till the temperature of the contents becomes \(15^{\circ} \mathrm{C}\). Calculate the amount of steam required, if water equivalent of the calorimeter and loss of heat due to radiation is negligible. | Latent heat of ice \(=80 \mathrm{cal} / \mathrm{gm}\) and I.atent heat of steam \(=536 \mathrm{cal} / \mathrm{gm}]\)

The gross radiation emitted by a perfectly black body is (a) dependent on its temperature (b) dependent on the area of its surface (c) dependent on the temperature of the surrounding (d) independent of the temperature of the surroundings
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