Unit Conversion

Problems

practice

1. A 20 km long, 8 m wide, two-lane highway is to be paved with a 4 cm thick layer of asphalt. A fleet of three dumptrucks is to be employed, each with an empty mass of 20 metric tons and a carrying capacity of 20 m³. Asphalt with a density of 0.72 g/cm³ will be used. Determine…
   1. the total volume of asphalt needed
   2. the number of dumptruck loads required
   3. the gross mass of each truck
2. Determine which of the following expressions are dimensionally consistent.

   1. v2 = 2ax

   2. x = vt + ½at3

   3. tan θ =	v
      	x

3. Write something else.
4. Write something completely different.

algebraic

• For each of the following problems let a [m/s²], t [s], v [m/s], and x [m].

1. Determine the SI unit of each of the following quantities.

   1. v2

      ax

   2. at2

      2

   3. 2π√	x
      	a

2. Determine which of the following expressions are dimensionally consistent.

   1. v = xt

   2. x = vt

   3. t = vx

   4. x = at + ½at2

   5. x = vt + ½at3

   6. v2 = 2ax2

   7. x =	√	2a
      		t

   8. t =	√	2x
      		a

   9. t =	√	v2
      		ax

   10. tan θ =	x
       	y

   11. tan θ =	y
       	x

   12. tan θ =	v
       	at

3. Determine the SI units of the constants C₁ and C₂ so that the following expressions are dimensionally correct.

   1. x = C1 + C2t

   2. x = C1t + ½C2t2

   3. v = C1t + C2t2

   4. x = C1 sin(2πC2t)

   5. v = ½C1e−C2t

4. In the formula below the symbol U is measured in kg m²/s² and x stands for length. What are the units of k?

   U = ½kx²

5. In the formula below the symbol T is measured in seconds and ℓ stands for length. What are the units of g?

   T = 2π√	ℓ
   	g

6. Derive the unit for the quantity P given the following equations…

   v =	x
   	t

   a =	v
   	t

   F = ma

   W = Fx

   P =	W
   	t

   with the following units…

   x	is in meters [m]
   t	is in seconds [s]
   m	is in kilograms [kg]

7. Derive the unit for the quantity g given the following equations…

   F = kx

   U = ½kx2

   U = mgh

   with the following units…

   F	is in newtons [kg m/s2]
   x, h	are in meters [m]
   m	is in kilograms [kg]