Laws of Motion Formulas

1. First law (law of inertia) (define force)

Without external force body does not change itΓÇÖs state of rest or motion.

2. Second law (measure force)

\(\overrightarrow{\mathrm{F}}=\frac{\mathrm{d}\overrightarrow{\mathrm{p}}}{\mathrm{dt}}\) = m\(\overrightarrow{\mathrm{a}}\)

3. Third law (gives direction of force)

ΓÇ£Every action has equal and opposite reactionΓÇ¥
\(\overrightarrow{\mathrm{F}}_{\mathrm{AB}}=-\overrightarrow{\mathrm{F}}_{\mathrm{BA}}\)

4. Impulse

\(\Delta \overrightarrow{\mathrm{P}}=\overrightarrow{\mathrm{P}}_{\mathrm{f}}-\overrightarrow{\mathrm{P}}_{\mathrm{i}}=\overrightarrow{\mathrm{F}} \times \Delta \mathrm{t}\)

5. Motion in a lift

Case 1
If the lift is uncelebrated \(\overrightarrow{\mathrm{a}}\) = 0 (\(\overrightarrow{\mathrm{v}}\) = constant or zero) Apparent weight = actual weight w’ = w = mg

Case 2

(a) If lift is accelerated upward (+ \(\overrightarrow{\mathrm{a}}\) = constant Γåæ)
w’ = mg + ma = m (g + a)
(b) If lift is accelerated upward with -ve acceleration
(- \(\overrightarrow{\mathrm{a}}\) = constant Γåæ)
w” = mg – ma = m(g – a)
If a = g, w’ = 0 (weightlessness)
If a > g then body comes in contact with ceiling of the lift.
w’ = – ve

6. Motion of a block on a horizontal smooth surface

Case 1

R = mg
F = ma
or a = \(\frac{F}{m}\)

Case 2

R = mg – F sin ╬╕
F cos ╬╕ = ma
or a = \(\frac{\mathrm{F} \cos \theta}{\mathrm{m}}\)

Case 3

R = mg + F sin ╬╕
a = \(\frac{\mathrm{F} \cos \theta}{\mathrm{m}}\)

7. Motion of bodies in contact

a = \(\frac{\mathrm{F}}{\mathrm{m}_{1}+\mathrm{m}_{2}+\mathrm{m}_{3}+\mathrm{m}_{4}}\)
If f₁ = contact force between masses m₁ and m₂
f₂ = contact force between masses m₂ and m₃
f₃ = contact force between masses m₃ and m₄
then F = (m₁ + m₂ + m₃ + m₄) a
f₁ = (m₂ + m₃ + m₄) a
f₂ = (m₃ + m₄) a
f₃ = m₄ a

8. Motion of connected bodies

a = \(\frac{\mathrm{F}}{\mathrm{m}_{1}+\mathrm{m}_{2}+\mathrm{m}_{3}+\mathrm{m}_{4}}\)
F = (m₁ + m₂ + m₃ + m₄) a
T₃ = (m₁ + m₂ + m₃) a
T₂ = (m₁ + m₂) a
T₁ = m₁ a

9. Motion of a body on a smooth inclined plane

R = mg cos ╬╕
ma = mg sin ╬╕
or a = g sin ╬╕
Special case:
When the smooth plane is moving horizontally with a acceleration (b) as shown in fig.

m (a + b cos ╬╕) = mg sin ╬╕ …(1)
m b sin ╬╕ = R – mg cos ╬╕ …(2)
solving above equation we get
a = g sin ╬╕ – b cos ╬╕
R = m (g cos ╬╕ + b sin ╬╕)
If a = 0 (means there is no relative motion between the blocks) then
b = g tan ╬╕
& If R = 0 (means body is released freely and all surfaces are smooth) then
b = – g cot ╬╕

10. Motion of two bodies connected by a string

Case 1

If m₂ > m₁ then
m₂ g – T = m₂ a …(1)
and T – m₁ g = m₁ a …(2)
solving equation (1) & (2)
a = \(\frac{\left(m_{2}-m_{1}\right)}{\left(m_{1}+m_{2}\right)}\).g
and T = \(\frac{2 m_{1} m_{2}}{m_{1}+m_{2}}\)g

Case 2

m₂g – T = m₂a …(1)
and T – m₁ g sin ╬╕ = m₁ a …(2)
solving equation (1) & (2)
a \(\frac{\left(m_{2}-m_{1} \sin \theta\right)}{\left(m_{1}+m_{2}\right)}\).g
and T = \(\frac{m_{1} m_{2} g}{m_{1}+m_{2}}\)(1 + sin ╬╕)

Case 3

m₁ g sin ╬▒ – T = m₁ a …(1)
and T – m₂ g sin ╬▓ = m₂ a …(2)
Solving equation (1) & (2)
a = \(\frac{g\left(m_{1} \sin \alpha-m_{2} \sin \beta\right)}{m_{1}+m_{2}}\)
T = \(\frac{m_{1} m_{2}}{m_{1}+m_{2}}\)(sin ╬▒ + sin ╬▓)g
Note:-
If whole system is moving

• Upwards
• Down wards
• Left or right,

with acceleration ΓÇ£aΓÇ¥ then replace ‘g’ with in

• (g + a), in
• (g – a) and in
• \(\sqrt{\mathrm{g}^{2}+\mathrm{a}^{2}}\), respectively in above formulas.

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