3.5 Problem Set

PROBLEM SET 1

1. Employ the dimensional set \(M\) (mass), \(V\) (velocity), \(T\) (time), and do the following exercise. The attached solutions may be consulted if necessary.

   1. The linear acceleration of a particle can be envisioned as the increase or decrease in the velocity of the particle, measured over a short time period. Write the dimensions of linear acceleration.
      
   2. The magnitude of force on a body wholly free to move can be calculated as the product of the body’s mass and its linear acceleration. Write the dimensions of mechanical force.
      
   3. The instantaneous kinetic energy of a moving body is proportional to the mass of the body and to the square of its linear speed. Write the dimensional formula for kinetic energy.

2. Employ the dimensional set \(M\) (mass), \(L\) (length), \(T\) (time), and do (a), (b), (c) of Exercise 1.

3. Employ the dimensional set \(F\) (force), \(L\) (length), \(T\) (time), and do the following exercise.

   1. Problem (c) of Exercise 1.
      
   2. The work done in moving the body of Problem 1(c) is equivalent to the product of the force acting on the body and the distance the body moves while the force is being applied. Write the dimensional formula for mechanical work.
      
   3. Power is defined as the time rate of energy expenditure (or, equivalently, as the rate at which work is done). Write the dimensional formula for the power of the agency that imparts the motion to the body of Problem 3(b).

4. Employ the dimensional set \(M\), \(L\), \(T\) and do (b), (c) of Exercise 3.

5. In the mks system of measurement, the meter (m) is the unit standard of length, the kilogram (kg) the unit standard of mass, and the second (sec) the unit standard of time. The derived unit of force in the mks system is called the “newton” (nt), the derived unit of energy is called the “joule”, and the derived unit of power is called the “watt”. Employ the mks system of measurement and write the unit magnitudes of the corresponding quantities described in Exercises 2, 3, 4.

PROBLEM SET 2

1. Determine the number of 24-hour days in the tropical year 1900.

2. Write algebraic formulae for the relationships between

   1. Kelvin and Fahrenheit temperature,
      
   2. Celsius and Fahrenheit temperature.

3. A molecule of water occupies a square cross-sectional area of about 10 \(\mathring{A}^2\). How many water molecules are needed to cover a square centimeter of surface?

4. Determine the kinetic energy in ergs of a molecule having a mass of \(2.0\times10^{-22}\)g and a velocity of \(4.0\times10^4\)cm\(\cdot\)sec^-1.

5. A red blood cell has a diameter of 7.5 microns; express its diameter in units of centimeter length.

6. Write as ordinary magnitudes of the units indicated:

   1. 3.5 milliliters of benzine
   2. 3.5 deciliters of seawater
   3. 1 kiloyear of Roman Empire
   4. 0.04 megawatts of electrical power
   5. 2 nanograms of a butterfly’s breakfast
   6. 0.002 nanograms of proton mass
   7. 200.5 picograms of hydronium ion
   8. 1.099 micromoles of H₂SO₄
   9. 1 hectare of rice paddy
   10. 10 square dekacentimeters of contiguous quadrat

7. Express the following quantities with appropriate metric prefixes:

   1. 7.35 \(\times\) 10⁹ liters
   2. 7.35 \(\times\) 10^-9 liters
   3. 1,000,000 watts
   4. 0.20 \(\times\) 10^-5 moles
   5. 8,575,000 microcuries
   6. 2.10 \(\times\) 10³ calories per gram
   7. 10^-3 photons/cm²

PROBLEM SET 3

1. Determine the dimensions of the following:
   (a)\(\int_{x_1}^{x_2}mv(x)dv\), where \(m\) is the mass of a body and \(v\) its velocity along a path from \(x_1\) to \(x_2\).
   (b)\(\frac{\partial}{\partial x}\int mvdv\)
   (c)Given \(R=\mu\int_xf(t)dt\), determine the dimensions of \(f(x)\) where \(R\) stands for frictional resistance, \(\mu\) is dynamic viscosity, and \(x\) is the Cartesian variable of location.

2. Given the Poisson equation \(\nabla^2\phi=f(x,y)\), where \(f(x,y)\) describes the distribution in a membrane of the ratio of load (force) per unit area to the tension (force) per unit length, determine the dimensions of the potential function \(\phi(x,y)\).

3. Given the equation of motion \(\ddot{x}+\omega^2x=f(t)\), determine the dimensions of \(\omega\) and \(f(t)\), where \(x\) is the measure of displacement.

4. Show that the equation \[-\rho\frac{\partial p}{\partial z}+\mu\frac{\partial^2\omega} {\partial z^2}=0\] is not correctly formulated, irrespective of its intended meaning, where \(\rho\) is fluid density, \(p\) is pressure, \(\mu\) is dynamic viscosity, \(\omega\) is fluid velocity, and \(z\) is the Cartesian dimension of displacement.

5. Employ the dimensions \(H\) (quantity of heat), \(\Theta\) (temperature), and \(M\) (mass).

   1. Write the dimensional formula for specific heat, which is defined as the quantity of heat required to impart a unit increase in temperature to a unit mass of substance.
      
   2. Formulate a scale unit of specific heat in terms of the gram, the degree Kelvin, and the calorie.

6. In many circumstances, the transfer of heat by conduction can be described by Fourier’s law \[\vec{q}=-k\nabla\theta\] where \(\vec{q}\) is the vector of heat current density (quantity of heat per unit area per unit time), \(\nabla\theta\) the temperature gradient vector, and \(k\) the coefficient of thermal conductivity (whose magnitude depends on the nature of the conducting substance). Expand the equation into its component parts and do the following problems.

   1. Employ the dimensional set {\(H\), \(\Theta\), \(L\), \(T\)} and write the dimensional formula for the coefficient of thermal conductivity. Define \(k\) in words.
   2. Employ the calorie, degree Celsius, centimeter, and second; write a scale unit for \(k\).
      
   3. Employ the joule, degree Kelvin, meter, and second; write a scale unit for \(k\).
      
   4. Employ the dimensional set {\(M\), \(\Theta\), \(L\), \(T\)} and write the dimensional formula for the mechanical equivalent of \(k\).
      
   5. Employ the gram, centimeter, degree Kelvin, and second; write a scale unit for \(k\).
      
   6. Employ the dimensional set {\(F\), \(\Theta\), \(L\), \(T\)} and write a dimensional formula for \(k\).
      
   7. Employ the newton, degree Kelvin, and second; write a scale unit for \(k\).
      
   8. Employ the watt, meter, and degree Kelvin; write a scale unit for k.
      
   9. Write a conversion factor between your units of (b) and (c).
      
   10. Write a conversion factor between your units of (c) and (e).

7. Determine the volume in cubic centimeters of a sphere of radius 1.5 ft.

8. The density of grain alcohol at 20\(^{\circ}\)C is 0.79 g\(\cdot\)cm^-3. Determine the mass of 30 ml of alcohol.

9. The blade of an ice skate makes contact with the ice over a length of about 15 cm and a width of 2.8 mm. Calculate the pressure on the ice produced by an ice-skater of 150 lbs mass. [The acceleration of gravity at the earth’s surface is about 980 cm\(\cdot\)sec^-2.]

10. A phonograph needle makes contact with a record surface over a circular area of diameter 80 \(\mu\)m. Calculate the pressure on the record when the needle arm weighs 2 ounces.