LIST OF FORMULAS:
1. Polar unit vectors in terms of cartesian unit vectors:
𝑟̂ (𝜃) = cos 𝜃𝑖̂ + sin 𝜃𝑗̂
𝜃̂(𝜃) = −𝑠𝑖𝑛 𝜃𝑖̂ + 𝑐𝑜𝑠 𝜃𝑗̂
2. Time derivative of the polar unit vectors:
ⅆ𝑟̂
= 𝜃̇ 𝜃̂,
ⅆ𝑡
ⅆ𝜃̂
= −𝜃̇ 𝑟̂
ⅆ𝑡
3. Velocity in polar coordinates:
𝑣⃗ = 𝑟̇ 𝑟̂ + 𝑟𝜃̇ 𝜃̂
4. Acceleration in polar coordinates:
𝑎⃗ = (𝑟̈ − 𝑟𝜃̇ 2 )𝑟̂ + (𝑟𝜃̈ + 2𝑟̇ 𝜃̇ )𝜃̂
5. Length element in plane polar:
ⅆ𝑟⃗ = ⅆ𝑟𝑟̂ + 𝑟ⅆ𝜃𝜃̂
6. Angular momentum (a.m.) for fixed axis rotation about z:
𝐿𝑍 = 𝐼0 𝜔 (𝑎. 𝑚. 𝑎𝑏𝑜𝑢𝑡 𝐶𝑀) + (𝑅⃗⃗ × 𝑀𝑣⃗)𝑍 (𝑎. 𝑚. 𝑜𝑓 𝑡ℎ𝑒 𝐶𝑀 𝑎𝑏𝑜𝑢𝑡 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑝𝑜𝑖𝑛𝑡)
7. For a gyroscope, the torque 𝜏⃗:
𝜏⃗ =
ⅆ𝐿⃗⃗
⃗⃗ × ⃗⃗⃗⃗
=𝛺
𝐿𝑠
ⅆ𝑡
⃗⃗ is the precessional angular velocity and ⃗⃗⃗⃗
𝛺
𝐿𝑠 is the spin angular momentum.
8. Moments and products of inertia:
𝐼𝑥𝑥 = ∑ 𝑚𝑖 (𝑦𝑖2 + 𝑧𝑖2 ) ,
𝑖
𝑖
𝐼𝑥𝑦 = − ∑ 𝑚𝑖 𝑥𝑖 𝑦𝑖 = 𝐼𝑦𝑥 ,
𝑖
𝐼𝑦𝑦 = ∑ 𝑚𝑖 (𝑧𝑖2 + 𝑥𝑖2 ), 𝐼𝑧𝑧 = ∑ 𝑚𝑖 (𝑦𝑖2 + 𝑥𝑖2 )
𝑖
𝐼𝑦𝑧 = − ∑ 𝑚𝑖 𝑦𝑖 𝑧𝑖 = 𝐼𝑧𝑦 , 𝐼𝑧𝑥 = − ∑ 𝑚𝑖 𝑧𝑖 𝑥𝑖 = 𝐼𝑥𝑧
𝑖
𝑖
9. Equation for a central force (with spherical symmetry):
𝐹⃗ (𝑟) = 𝑓(𝑟)𝑟̂
10. Equation for reduced mass (𝜇 ) picture in a two-body problem:
𝜇𝑟⃗̈ = 𝑓(𝑟)𝑟̂
11. Energy (𝐸) of a particle of mass 𝑚 moving under a central force:
1
𝐿2
𝐸 = 𝑚𝑟̇ 2 + 𝑈𝑒𝑓𝑓 𝑤ℎ𝑒𝑟𝑒 𝑈𝑒𝑓𝑓 =
+ 𝑈(𝑟)
2
2𝑚𝑟 2
Angular momentum L= 𝑚𝑟 2 𝜃̇ and 𝑈(𝑟) is the potential energy.
12. Orbit equation for an object moving under a central force field:
𝑟
ⅆ𝑟
𝜃 − 𝜃0 = 𝐿 ∫
𝑟0
𝑟 2 √2𝜇(𝐸 − 𝑈𝑒𝑓𝑓 )
13. Taylor series expansion of the potential energy, assuming small perturbations:
1
ⅆ2 𝑈
ⅆ2 𝑈
𝑈(𝑟) ≈ 𝑈(𝑟0 ) + (𝑟 − 𝑟0 )2 2 ⃒𝑟0 anⅆ effective spring constant 𝑘 =
⃒
2
ⅆ𝑟
ⅆ𝑟 2 𝑟0
Where 𝑈(𝑟) = 𝑈(𝑟)𝑚𝑖𝑛 at 𝑟 = 𝑟0 .
14. Equation of orbit of an object moving under gravity described by a conic section in polar
coordinates:
𝑟=
𝑟0
𝐿2
2𝐸𝐿2
𝑤ℎ𝑒𝑟𝑒 𝑟0 ≡
𝑎𝑛ⅆ 𝜖 ≡ √1 +
1 − 𝜖𝑐𝑜𝑠𝜃
𝜇𝐶
𝜇𝐶 2
Total energy 𝐸, eccentricity is 𝜖 and 𝐶 = 𝐺𝑚1 𝑚2 where 𝑚1 𝑎𝑛ⅆ 𝑚2 are the two masses and
gravitational constant 𝐺 = 6.67 × 10−11 𝑁𝑚2 /𝑘𝑔2, 𝐿 is the angular momentum.
15. Length of major axis (𝐴) and minor axis (𝐵) of an elliptical orbit:
𝐴 = 𝑟𝑚𝑎𝑥 + 𝑟𝑚𝑖𝑛 = 2𝑎 =
𝐵 = 2𝑏 =
2𝑟0
√1 − 𝜖 2
2𝑟0
𝐶
=
1 − 𝜖 2 −𝐸
=
𝐿
√−2𝜇𝐸
16. Velocity 𝑣 at any point 𝑟 of the elliptical orbit:
𝑣2 =
2𝐶 1 1
( − )
𝜇 𝑟 𝐴
17. The time period of revolution 𝑇 of an object in an elliptical orbit as given by Kepler’s 3rd law:
𝑇2 =
𝜋2𝜇 3
𝐴
2𝐶
where 𝐴 is the major axis length.
𝑑𝑈
18. In one dimension, for a conservative force, 𝐹 = − 𝑑𝑟 where 𝑈 is the potential energy.