1.1.3 Estimating Physical Quantities

• When a number is expressed in an order of 10, this is an order of magnitude.
  • Example: If a number is described as 3 × 10⁸ then that number is actually 3 × 100 000 000
  • The order of magnitude of 3 × 10⁸ is just 10⁸

• Orders of magnitude follows rules for rounding
  • The order of magnitude of 6 × 10⁸ is 10⁹ as the magnitude is rounded up

• A quantity is an order of magnitude larger than another quantity if it is about ten times larger
• Similarly, two orders of magnitude would be 100 times larger, or 10²
  • In physics, orders of magnitude can be very large or very small

• When estimating values, it’s best to give the estimate of an order of magnitude to the nearest power of 10
  • For example, the diameter of the Milky Way is approximately 1 000 000 000 000 000 000 000 m

• It is inconvenient to write this many zeros, so it’s best to use scientific notation as follows:

1 000 000 000 000 000 000 000 = 1 × 10²¹ m

• The order of magnitude is 10²¹
• Orders of magnitude make it easier to compare the relative sizes of objects
  • For example, a quantity with an order of magnitude of 10⁶ is 10 000 times larger than a quantity with a magnitude of 10²

Estimating Physical Quantities Table

• To estimate is to obtain an approximate value
  • For very large or small quantities, using orders of magnitudes to estimate calculations is a valid approach

• Estimation is typically done to the nearest order of magnitude

Part (a)

The temperature of the oven:

Step 1: Identify the approximate temperature of an oven

• • • A conventional oven works at ∼200 °C which is 473 K

Step 2: Identify the order of magnitude

• • • Since this could be written as 4.73 × 10² K
    • The order of magnitude is ∼10²

Part (b)

The volume of the Earth (in m³):

Step 1: Identify the approximate radius of the Earth

• • • The radius of the Earth is ∼6.4 × 10⁶ m

Step 2: Use the radius to calculate the volume

• • • The volume of a sphere is equal to:

V = 4/3 π r³

V = 4/3 × π × (6.4 × 10⁶)³

V = 1.1 × 10²¹ m³

Step 3: Identify the order of magnitude

• • • The order of magnitude is ∼10²¹

Part (c)

The number of seconds in 95 years:

Step 1: Find the number of seconds in a single year

1 year = 365 days with 24 hours each with 60 minutes with 60 seconds

365 × 24 × 60 × 60 = 31 536 000 seconds in a year

Step 2: Find the number of seconds in 95 years

95 × 31 536 000 = 283 824 000 seconds

• • • This is approximately 2.84 × 10⁸ seconds
    • Therefore the order of magnitude is ∼10⁸