7 Equilibrium & Newton’s First Law
Key Takeaways
- Extra Help: Newton’s First Law
- Extra Help: Equilibrium and Statics
Equations Introduced and Used for this Topic: (All equations can be written and solved as both scalar and vector and all equations are generally solved as vectors)
- [latex]\sum F_{x, y, z} = 0 \text{ N}[/latex]
- [latex]T = r \times F[/latex]
- [latex]T_{\text{clockwise}} = T_{\text{counter-clockwise}}[/latex]
- [latex](m_1 + m_2) x_{\text{cm}} = m_1 x_1 + m_2 x_2[/latex]
Where…
- [latex]T[/latex] is the torque, measured in newton-metres (Nm)
- [latex]F[/latex] is force, measured in newtons (N)
- [latex]r[/latex] is the length of the lever-arm measured in metres (m)
- [latex]m_1[/latex] and [latex]m_2[/latex] are the masses of two bodies, measured in kilograms (kg)
- [latex]x_{\text{cm}}[/latex], [latex]x_1[/latex], and [latex]x_2[/latex] are the distances, measured in metres (m)
- ø is the angle given for the triangle and is generally between the adjacent side and the hypotenuse with the angle measured in degrees (be careful to make sure that the calculator is not working in gradients or radians).
- Hypotenuse is the longest side in a right triangle and is opposite to the 90° angle.
- Opposite is the side that is opposite to the angle that you are given.
- Adjacent is the side that is beside the angle you are given.
The Trigonometric Functions:
- [latex]\text{sine ø } = \dfrac{\text{opposite}}{\text{hypotenuse}}[/latex]
- [latex]\text{cosine ø } = \dfrac{\text{adjacent}}{\text{hypotenuse}}[/latex]
- [latex]\text{tangent ø } = \dfrac{\text{opposite}}{\text{adjacent}}[/latex]
7.1 Right Angle Trigonometric Functions
Trigonometry, a term derived from the Greek trigonon (triangle) and metron (measure) has known historical roots dating back 4000 years to writings found in Egyptian and Babylonian mathematics and astronomical records. Indian astronomers were using trigonometry over 2600 years ago and that widespread usage by Islamic and Chinese scientists was common at least 1500 years prior.
The study of trigonometry returned to European nations in the Renaissance in translations of Arabic and Greek writings. Modern trigonometry is credited as reaching its current form through the works of Leonard Euler (1707-1783), who is considered to be one of the greatest mathematicians in history and had worked in nearly every area of mathematics and physics. Modern notation used in trigonometry comes directly from Euler’s writings, as are many other currently used notations in mathematics and physics. A list of Euler’s achievements is extensive as well as his writings that have been collected in 92 volumes, making him one of the most prolific writers in mathematics history.
Introductory trigonometry is based on identifying the similarities between identical right angled (one angle is 90°) of different sizes. If the angles of a triangle are identical then all of these triangles are simply larger or smaller copies of each other. In all of the cas