The Equations of Curved Mirrors and Lenses
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|G=\displaystyle \frac {h_{i}}{h_{o}} = \frac {-d_{i}}{d_{o}} = \frac {-l_{f}}{l_{o}} = \frac {-l_{i}}{l_{f}}|
|\displaystyle \frac {1}{d_{o}} + \frac {1}{d_{i}} = \frac {1}{l_{f}}|
|{d_{i}} = {l_{i}} + {l_{f}}|
|{d_{o}} = {l_{o}} + {l_{f}}|
|{l_{i}} \times {l_{o}} = {l_{f}}^2|
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|l_{f}|: focal length (or focal distance)
|d_{o}|: object-mirror distance |d_{i}|: image-mirror distance |l_{o}|: object-focus distance |l_{i}|: distance image-focus |h_{o}|: pitch of the object |h_{i}|: height of the image |R|: radius of curvature |
| |R = 2 \times {l_{f}}| | (only in curved mirrors) |
Signs Convention for Mirrors
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Measurement |
Positive Sign |
Negative Sign |
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Mirror-Image Distance (|d_{i}|) |
The image is real. | The image is virtual. |
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Focal Length (|l_{f}|) |
The mirror is concave (convergent). | The mirror is convex (divergent). |
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Magnification (|G|)
Image Height (|h_{i}|)
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The image is upright. | The image is upside down. |
Signs Convention for Lenses
| Measurement | Positive Sign | Negative Sign |
| Mirror-Image Distance (|d_{i}|) | The image is real (on the opposite side of the lense to the object). | The image is virtual (on the same side of the lense as the object). |
| Focal Length (|l_{f}|) | The lense is convex (convergent). | The lense is concave (divergent). |
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Magnification (|G|)
Image Height (|h_{i}|)
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The image is upright. | The image is upside down. |
Refraction
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|n_{1}\times \sin \theta_{i} = n_{2}\times \sin\theta_{r}| |
|n_{1}|: refractive index of environment 1 |
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|C = \displaystyle \frac {1}{l_{f}}| |
|C|: vergence of the lense |( \delta )| |
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|C = (n - 1) \times \displaystyle (\frac {1}{R_{1}} - \frac {1}{R_{2}})| |
|C| : vergence of the lense |( \delta )| |
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| |C_T=C_1+C_2+...+C_n| |
|C_T| : total vergence |(\delta)|
|C_1|,|C_2|,|C_n|: individual vergence of each lense |(\delta)| |
The Equations of Uniformly Accelerated Rectilinear Motion (UARM)
| |v_{moy}=\displaystyle \frac{\triangle x}{\triangle t}| |a=\displaystyle \frac{\triangle v}{\triangle t}| |v_{f}=v_{i} + a \cdot {\triangle t}| |\triangle x= \displaystyle \frac{(v_{i} + v_{f}) \cdot {\triangle t}}{2}| |\triangle x= v_{i} \cdot \triangle t +\displaystyle \frac{1}{2} \cdot a \cdot {\triangle t}^{2}| |{v_{f}}^2={v_{i}}^2+2 \cdot a \cdot \triangle x| |
|\triangle x = x_{f} - x_{i}|: variation in position |\text {(m)}| |v_{moy}|: average speed |\text {(m/s)}| |v_{i}|: initial velocity |\text {(m/s)}| |v_{f}|: final velocity |\text {(m/s)}| |a|: acceleration |\text {(m/s}^2)| |\triangle t = t_{f} - t_{i}|: time variation |\text {(s)}| |
Kinematic
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The Range |
|\text{Range} = \displaystyle \frac{v_i^2 \, sin\, 2 \theta _i}{g}| |
|\text{Range}| : range |\text{(m)}|
|v_i| : initial velocity |\text{(m/s)}| |\theta _i|: intinial angle from the horizontal |(^{\circ})| |g|: gravitational acceleration |\text{(m/s}^2)| |
Dynamic
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|a = g \times \sin \theta| |
|a|: acceleration |\text {(m/s}^2)| |
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|\displaystyle g = \frac{G \cdot m}{r^{2}}| |
|g|: gravitational acceleration |\text {(m/s}^2)| |G|: universal gravitation constant |\left(6.67 \times 10^{-11} \displaystyle \frac {\text {N} \cdot \text{m}^{2}}{\text{kg}^{2}}\right)| |m|: mass of the celestial body |\text {(kg)}| |r|: celestial body radius |\text {(m)}| |
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|F_{f} = F_{m} - F_{R}| |
|F_{f}|: friction force |\text {(N)}| |
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| |F_{f} = \mu \cdot F_{N}| | |F_{f}|: friction force |\text {(N)}| |\mu|: coefficient of friction |F_{N}|: normal force |\text {(N)}| |
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|F_ {R} = m \times a| |
|F_{R}|: resultant force |\text {(N)}| |
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| |F_{g} = m \times g| |
|F_{g}|: gravitational force |\text {(N)}| |
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| |\displaystyle F_{g} = \frac{G \cdot m_{1} \cdot m_{2}}{r^{2}}| | |F_{g}|: force of attraction between celestial bodies |\text {(N)}| |G|: universal gravitation constant |\left(6.67 \times 10^{-11} \displaystyle \frac {\text {N} \cdot \text{m}^{2}}{\text{kg}^{2}}\right)| |m_{1}|: mass of the first object |\text {(kg)}| |m_{2}|: mass of the second object |\text {(kg)}| |r|: distance separating the two objects |\text {(m)}| |
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The Centripetal Acceleration |(a_c)| and the Centripetal Force |(F_c)| |
|a_{c} = \displaystyle \frac {v^{2}}{r}|
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|a_{c}|: centripetal acceleration |\text {(m/s}^2)| |v|: object's rotational speed |\text {(m/s})| |r|: radius of the circle |\text {(m)}| |
| |F_{c} = m \times \displaystyle \frac {v^{2}}{r}| | |F_{c}|: centripetal force |\text {(N)}| |m|: mass |\text {(kg)}| |v|: object's rotational speed |\text {(m/s)}| |r|: radius of the circle |\text {(m)}| |
Transformation of Energy
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|W = F \times \triangle x| |
|W|: work |\text {(J)}| |
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|P = \displaystyle \frac {W}{\triangle t}| |
|P|: mechanical power |\text {(W)}| |W|: work |\text {(J)}| |\triangle t|: variation in time |\text {(s)}| |
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|E_{p_{g}} = m \times g \times \triangle y| |
|E_{p}|: gravitational potential energy |\text {(J)}| |m|: mass |\text {(kg)}| |g|: gravitational field intensity |\text {(m/s}^2)| |\triangle y|: distance travelled (height) of the object |\text {(m)}| |
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| |E_{p_{e}} = \displaystyle \frac {1}{2} \times k \times \triangle x^{2}| | |E_{p_{e}}|: elastic potential energy |\text {(J)}| |k|: returning spring constant |\text {(N/m)}| |\triangle x|: distance travelled by the spring |\text {(m)}| |
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| |E_{k} = \displaystyle \frac {1}{2} \times m \times v^{2}| | |E_{k}|: kinetic energy |\text {(J)}| |m|: mass of the object |\text {(kg)}| |v|: velocity of the object |\text {(m/s)}| |
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| |E_m = E_k + E_p| | |E_{m}|: mechanical energy |\text {(J)}| |E_{p}|: potential energy |\text {(J)}| |E_{k}|: kinetic energy |\text {(J)}| |
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|F_{rappel} = - k \times \triangle x| |
|F_{rappel}|: returning force |\text {(N)}|
|k|: spring constant |\text {(N/m)}| |\triangle x|: spring deformation or compression |\text {(m)}| |
| | In parallel: |k_{eq} = k_1 + k_2| In series: |\displaystyle \frac {1}{k_{eq}} = \displaystyle \frac {1}{k_1} + \displaystyle \frac {1}{k_2}| |
|k_{eq}| : equivalent returning spring constant |\text {(N/m)}|
|k_1|,|k_2| : returning spring constant of each spring |\text {(N/m)}| |