Units and Measurements - Key Formulas

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1. Plane Angle

Formula:

What each symbol stands for:

• $d\theta the$ 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 $ds=1.0$ m.

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2. Solid Angle

Formula:

What each symbol stands for:

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

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

• $r$: radius of the sphere

Example 

On a sphere of radius $r=1.0$ m, a surface patch has area $dA=0.20$ m².

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3. Density

Formula:

What each symbol stands for:

• $\rho$: mass density

• $m$: mass of the object

• $V$: volume occupied by the object

Example 

A block of mass $m=12.0$ g occupies volume $V=3.0$ cm³.

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4. Newton’s Second Law (Force)

Formula
[ F = m,a ]
What each symbol stands for

• (F): net force acting on an object
• (m): 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\quad\text{(N)} ]

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5. Kinematic Equation (Constant Acceleration)

Formula
[ x = x_0 + v_0,t + \tfrac12,a,t^2 ]
What each symbol stands for

• (x): displacement after time (t)
• (x_0): initial displacement (at (t=0))
• (v_0): initial velocity (at (t=0))
• (t): elapsed time
• (a): constant acceleration

Simple example
Start from (x_0=0), with (v_0=2.0) m/s and (a=1.0) m/s². After (t=3.0) s:
[ x = 0 + (2.0)(3.0) + \tfrac12(1.0)(3.0)^2 = 6.0 + 4.5 = 10.5\quad\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\quad\text{s} ]

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7. Percentage (Relative) Error

Formula
[ %;\text{error} = \frac{\Delta x}{x}\times100 ]
What each symbol stands for

• (%;\text{error}): percentage uncertainty in the measurement
• (\Delta x): absolute error in the measured quantity
• (x): measured value

Simple example
A length (x=50.0) cm has an uncertainty (\Delta x=0.5) cm.
[ %;\text{error} = \frac{0.5}{50.0}\times100 = 1.0% ]

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8. Error in a Product

Formula
[ \Delta(AB) = A,B;\bigl(\tfrac{\Delta A}{A} + \tfrac{\Delta B}{B}\bigr) ]
What each symbol stands for

• (\Delta(AB)): absolute error in the product (A \times B)
• (A, B): measured quantities being multiplied
• (\Delta A, \Delta B): absolute errors in (A) and (B)

Simple example
(A=4.0\pm0.1), (B=2.0\pm0.2):

• Product: (4.0\times2.0=8.0)
• Relative errors: (\tfrac{0.1}{4.0}=0.025,;\tfrac{0.2}{2.0}=0.10)
• Sum of rel. errors: (0.025+0.10=0.125)
• Absolute error: (\Delta(AB)=8.0\times0.125=1.0)
  [ AB = 8.0 \pm 1.0 ]

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With these, you have each formula, what every symbol represents, and a straightforward example showing its use.