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1.7 Dimensions

Physical science seems to contain a bewildering array of different things that can be measured and quantified — mass, density, temperature, power, force, acceleration, and so on. Yet almost every physical quantity is a combination of only three fundamental quantities, or dimensions: mass, length, and time. These dimensions are abbreviated as M, L, and T.

The best way to see that all measurements can be reduced to a combination of three fundamental quantities is to consider some examples.

• Area of a square A = d2 has dimensions of L2

Visual representation of cube volume. Click here for original source URL.

• Volume of a cube V = d3 has dimensions of L3

• Density r = M / V has dimensions of M / L3

• Velocity v = d / t has dimensions of L / T

• Acceleration a = v / t has dimensions of L / T2

• Momentum S = mv has dimensions of M L / T

• Force F = ma has dimensions of M L / T2

• Pressure P = F / A has dimensions of M / L T2

• Kinetic energy E = 1/2 mv2 has dimensions of M L2 / T2

• Power P = E / t has dimensions of M L2 / T3

You can always check that an equation in physics makes sense by inserting the dimensions of the quantities, and demonstrating that the dimensions of the left-hand side and the right-hand side are the same. For example, we see that the dimensions of force are mass times length divided by time squared by starting with the equation F = ma. Substituting for acceleration we get F = mv/t. Substituting for velocity we get F = md/t2, dimensions of ML/T2. You will notice in algebra that we often omit the times sign (×), since it can be confused with the symbol x. When two numbers or symbols appear next to each other, you can assume that they are multiplied together.

Universal gravitational constant (G). Click here for original source URL.

What are the dimensions of the gravitational constant? Newton's law of gravity is written:

F = G m1 m2 / r2

In this equation, m1 and m2 are the masses of any two objects and r is the distance between them. We can rewrite the equation using the dimensions of each quantity:

M L / T2 = G (M M / L2)

To get G on its own, we follow the rules of algebra. Multiplying each side of the equation by L2 gives:

M L3 / T2 = G M2

Now dividing the each side of the equation by M2 gives the result:

The age of the universe can be determined by measuring the Hubble constant?today and extrapolating back in time with the observed value of density parameters (?). Before the discovery of?dark energy, it was believed that the universe was matter-dominated, and so ? on this graph corresponds to ?m. Note that the?accelerating universe?has the greatest age, while theBig Crunch?universe has the least age. Click here for original source URL.

L3 / T2 M = G

So the dimensions of the gravitational constant G are length cubed divided by time squared times mass. Here is another example. The Hubble relation says that the recession velocity of a distant galaxy is proportional to its distance from us. In equation form, v = H0d, where H0 is the Hubble constant. Dividing each side of the equation by d, we get v/d = H0. Therefore, the dimensions of the Hubble constant are (L/T)/L or 1/T. Taking the reciprocal, the dimensions of 1/H0 are T, or time. The reciprocal of the Hubble constant is a measure of the age of the universe.