UNIT 1: Units and Measurements - Key Formulas

Units and Measurements - Key Formulas

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1. Plane Angle

 

Formula:$$d\theta =\frac{ds}{r}$$

What each symbol stands for:

• $d\theta$: infinitesimal plane angle

• $ds$: infinitesimal arc length

• $r$: radius of the circle

Example A circle of radius $r=2.0$ m has an arc length $ds=1.0ds=1.0$ m.

$$d\theta =\frac{1.0}{2.0}=0.50\text{ rad}$$

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2. Solid Angle

 

Formula

$$d\mathrm{\Omega }=\frac{dA}{r^2}$$

What each symbol stands for

• $d\mathrm{\Omega }$: infinitesimal solid angle

• $dA$: small patch of area on a spherical surface

• $r$: radius of the sphere

Simple example On a sphere of radius $r=1.0$ m, a tiny patch has $dA=0.20$ m².

$$d\mathrm{\Omega }=\frac{0.20}{{1.0}^2}=0.20\text{(steradian)}$$

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3. Density

 

Formula

$$\rho =\frac{m}{V}$$

What each symbol stands for

• $\rho$: mass density of the material

• $m$: mass of the object

• $VV$ volume occupied by the object

Simple example A metal block has mass $m=12.0$ g and volume$V=3.0$ cm³.

$$\rho =\frac{12.0}{3.0}=4.0\text{(g/cm³)}$$

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4. Newton’s Second Law (Force)

 

Formula

$$F=ma$$

What each symbol stands for

• $F$: net force acting on an object

• $mm$: mass of the object

• $a$: acceleration of the object

Simple example A cart of mass $m=2.0$ kg accelerates at $a=3.0$ m/s².

$$F=2.0\times 3.0=6.0\text{(N)}$$

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5. Kinematic Equation (Constant Acceleration)

 

Formula

$$x=x_0+v_{0}t+\frac{1}{2}a{t}^2$$

What each symbol stands for

• $x$: displacement after time $tt$

• $x_0$: initial displacement (at $t=0t=0$)

• $v_0$: initial velocity (at $t=0t=0$)

• $t$: elapsed time

• $a$: constant acceleration

Simple example Start from $x_0=0$, with $v_0=2.0$ m/s and $aa=1.0$ m/s². After $t=3.0$ s:

$$x=0+(2.0)(3.0)+\frac{1}{2}(1.0)(3.0{)}^2=6.0+4.5=10.5\text{m}$$

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6. Period of a Simple Pendulum

 

Formula

$$T=2\pi \sqrt{\frac{l}{g}}$$

What each symbol stands for

• $T$: time period of one complete oscillation

• $\pi$: mathematical constant (≈ 3.1416)

• $l$: length of the pendulum

• $g$: acceleration due to gravity

Simple example A pendulum of length $l=1.0$ m on Earth ($g=9.8$ m/s²):

$$T=2\pi \sqrt{\frac{1.0}{9.8}}\approx 2.01\text{s}$$

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7. Percentage (Relative) Error

 

Formula

$$\mathrm{\%}\text{error}=\frac{\mathrm{\Delta }x}{x}\times 100$$

What each symbol stands for

• $\mathrm{\%}\text{error}$: percentage uncertainty in the measurement

• $\mathrm{\Delta }x$: absolute error in the measured quantity

• $x$: measured value

Simple example A length $x=50.0$ cm has an uncertainty $\mathrm{\Delta }x=0.5$ cm.

$$\mathrm{\%}\text{error}=\frac{0.5}{50.0}\times 100=1.0\mathrm{\%}$$

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8. Error in a Product

 

Formula

$$\mathrm{\Delta }(AB)=AB(\frac{\mathrm{\Delta }A}{A}+\frac{\mathrm{\Delta }B}{B})$$

What each symbol stands for

• $\mathrm{\Delta }(AB)$: absolute error in the product $A\times BA\; \backslash times\; B$

• $A,B$: measured quantities being multiplied

• $\mathrm{\Delta }A,\mathrm{\Delta }B\backslash$: absolute errors in $A$ and $B$

Simple example $A=4.0\pm 0.1$, $B=2.0\pm 0.2$:

• Product: $4.0\times 2.0=8.04.0\backslash times2.0=8.0$

• Relative errors: $\frac{0.1}{4.0}=0.025,\frac{0.2}{2.0}=0.10$

• Sum of rel. errors: $0.025+0.10=0.125$

• Absolute error: $\mathrm{\Delta }(AB)=8.0\times 0.125=1.0$

$$AB=8.0\pm 1.0$$