Atwood machine

An Atwood machine is a simple mechanical system used in physics to study the principles of classical mechanics. It consists of a pulley with two masses hanging on either side of the pulley, connected by a string that runs over the pulley. By analyzing the Atwood machine, one can understand concepts such as tension, acceleration, and the relationship between mass and force. This setup allows for the application of Newton's laws of motion to describe the motion of the masses and the pulley.

Masses

Mass 1 Mass 2

Position of masses

Mass 1 Mass 2

Initial velocity of the masses

u For

m1 m2

World

Gravity

All inputs in SI units.
Rope and pully are massless.
+ direction downwards.

Theory

A Atwood Machine consists of 2 masses connected through a rope that passes over a pulley. Each mass has 2 forces acting on it - the force of gravity and the tension of the string. These 2 forces act on the opposite direction, gravity tries to pull the mass down, the tension force always tries to pull the mass up (remember: tension doesn't push)

Since a massless and inextensible string is assumed, the tension at all points on the rope remains constant. Also, since the length of the string will be constant, both the masses will have equal and opposite acceleration and velocity at all points.

Lets assume the acceleration for ${m_1}$ is downwards, then the acceleration for ${m_2}$ will be upwards. The equations of motion will be:

m_1a = m_1g - T \qquad \qquad 1 \cdot a = 1 \cdot 9.8 - T\newline m_2a = T - m_2g \qquad \qquad 2 \cdot a = T - 2\cdot 9.8

Solving for T,

T = \frac{2 m_1 m_2 g}{m_1 + m_2} \qquad \qquad T = \frac{2 \cdot 1 \cdot 2 \cdot 9.8}{1 + 2} = 13.07N

Using

{T}

in the equation for

{m_1}

a = \frac{m_1 g - T}{m_1} \qquad \qquad a = \frac{1\cdot 9.8 - 13.07}{1} = -3.267m/s^2

If ${a = 0}$ , the system may be at rest or moving with constant velocity (if the initial velocity ${u \ne 0}$ )

Tension: 13.07 N

Acceleration for m1: -3.267 m/s²

Free Body Diagram (FBD)

13.07 ↑

←0.000

0.000 →

9.800 ↓

13.07 ↑

←0.000

0.000 →

19.60 ↓

Visualisation

Time: 0.000 s

Velocity of m_1: 0.000 m/s