## Pulley Mass System | Definition, Examples in Physics – Laws of Motion

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**Pulley Mass System Physics:
**Determine the acceleration of the masses and the tension in the string.

(i) When unequal masses m_{1} and m_{2} are suspended from a pulley (m_{1} > m_{2})

m_{1}g – T = m_{1}a, and T – m_{2}g = m_{2}a

On solving equations, we get

\(a=\frac{\left(m_{1}-m_{2}\right)}{\left(m_{1}+m_{2}\right)} g,\)

\(T=\frac{2 m_{1} m_{2}}{\left(m_{1}+m_{2}\right)} g\)

(ii) When a body of mass m_{2} is placed on a frictionless horizontal surface, then

Mass Pulley System acceleration, a = \(\frac{m_{1} g}{\left(m_{1}+m_{2}\right)}\)

Tension in a Pulley System with Two Masses, T = \(\frac{m_{1} m_{2} g}{\left(m_{1}+m_{2}\right)}\)

(iii) When a body of mass m_{2} is placed on a rough horizontal surface, then

Acceleration, a = \(\frac{\left(m_{1}-\mu m_{2}\right) g}{\left(m_{1}+m_{2}\right)}\)

Tension in string, T = \(\frac{m_{1} m_{2}(1+\mu) g}{\left(m_{1}+m_{2}\right)}\)

(iv) When two masses m_{1} and m_{2} (m_{1} > m_{2}) are connected to a single mass M as shown in figure, then

m_{1}g – T_{1} = m_{1}a ……(i)

T_{2} – m_{2}g = m_{2}a ……(ii)

T_{1} – T_{2} = Ma ……(iii)

Acceleration, a = \(\frac{\left(m_{1}-m_{2}\right) g}{\left(m_{1}+m_{2}+M\right)}\)

Tension, T_{1} = \(\left(\frac{2 m_{2}+M}{m_{1}+m_{2}+M}\right) m_{1} g\)

T_{2} = \(\left(\frac{2 m_{1}+M}{m_{1}+m_{2}+M}\right) m_{2} g\)

(v) Motion on a smooth inclined plane, then

m_{1}g – T = m_{1}a ……(i)

T – m_{2}g sin θ = m_{2}a ……(ii)

Acceleration, a = \(\left(\frac{m_{1}-m_{2} \sin \theta}{m_{1}+m_{2}}\right) g\)

Tension, T = \(\frac{m_{1} m_{2}(1+\sin \theta) g}{\left(m_{1}+m_{2}\right)}\)

(vi) Motion of two bodies placed on two inclined planes having different angle of inclination, then

Acceleration, a = \(\frac{\left(m_{1} \sin \theta_{1}-m_{2} \sin \theta_{2}\right) g}{m_{1}+m_{2}}\)

Tension, T = \(\frac{m_{1} m_{2}}{m_{1}+m_{2}}\left(\sin \theta_{1}+\sin \theta_{2}\right) g\)

**Laws of Motion:
**There are various laws in Physics that define the motion of the object. When an object is in motion whether it is linear or circular there is some force which is always imposed on it.