Physics · Mechanics · Grade 10

Inclined plane and Atwood machine

Why is it easier to drag a load up a ramp, yet the work does not decrease?

The situation

It is hard to lift a heavy crate onto a lorry by hand, so workers put up a ramp and drag it up. The longer the ramp, the easier the pull — but does the work done decrease too? At what slope does the crate start sliding by itself, and in what order do a tube and cylinders roll down a ramp? In the laboratory we also test Newton's laws with an Atwood machine and a pulley.

Topic

Inclined plane and Atwood machine. Loading a lorry along a ramp: friction, pulley and efficiency

Key ideas

Inclined plane
Force of friction
Efficiency

Formulas

F_{∥} = m g \sin θ, N = m g \cos θ, F_{f r} \leq μ_{0} N

We resolve the weight into components along the inclined plane and perpendicular to it; when sliding, $F_{f r} = μ N$ .

tg θ_{0} = μ_{0}, a = g (\sin θ - μ \cos θ)

The sliding threshold and the acceleration when sliding down (when pushed upwards, $a = - g (\sin θ + μ \cos θ)$ ).

η = \frac{A_{f}}{A_{t}} = \frac{m g h}{F l} = \frac{\sin θ}{\sin θ + μ \cos θ}

Efficiency of an inclined plane: useful work (lifting) / total work (pulling).

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

Atwood machine (string and pulley massless). With the pulley: $a = (m_{2} g - m_{1} g \sin θ \mp μ m_{1} g \cos θ) / (m_{1} + m_{2})$ .

a = \frac{g \sin θ}{1 + I / (m r^{2})}

Acceleration of a body rolling without slipping: part of the energy goes into rotation.

Theory

Resolving the forces along the inclined plane

The forces on the crate are its weight $m \vec{g}$ , the normal reaction $\vec{N}$ , friction and the pulling force. We choose axes along the inclined plane and perpendicular to it: the component of $m g$ along the plane, $m g \sin θ$ , pulls the crate down, and the perpendicular component $m g \cos θ$ is balanced by the support ( $N = m g \cos θ$ ). Static friction can take any value up to $μ_{0} N$ , while kinetic friction equals $μ N$ and points against the motion ( $μ < μ_{0}$ ).

The sliding threshold and events

A crate that is not pulled starts to slide when $m g \sin θ > μ_{0} m g \cos θ$ , that is, $\tan θ > μ_{0}$ — the mass cancels. Sliding down, $a = g (\sin θ - μ \cos θ)$ . A crate pushed upwards is slowed by weight and friction together, so it stops with $a = - g (\sin θ + μ \cos θ)$ , and then, if $\tan θ > μ_{0}$ , slides back down with a different (smaller) acceleration. The simulation calculates the motion in pieces split exactly at these events (setting off, stopping, reaching the end). When pulling, a larger force is needed at the start — the “peak” on the force meter graph is the moment static friction is overcome.

Lab work: efficiency of an inclined plane

Use a force meter to measure the weight $m g$ of the crate, then pull it steadily along the ramp and record the force meter reading $F$ ; measure the ramp length $l$ and height $h$ . Useful work $A_{f} = m g h$ (raising the load), total work $A_{t} = F l$ , $η = A_{f} / A_{t} \cdot 100 %$ . Because of friction $η < 1$ ; as the slope increases, the efficiency rises, but the pulling force rises too. Without friction $F l = m g h$ : an inclined plane gives a gain in force, not in work (the golden rule of mechanics). The table of results and the $η (θ)$ graph fill in after each pull.

Lab work: the Atwood machine

Two equal masses $M$ hang from a string passed over a pulley, and a small extra mass $m$ is placed on one of them. The system moves with uniform acceleration $a = m g / (2 M + m)$ . Measuring the time $t_{1}$ to cover the distance $h_{1}$ , we find $a = 2 h_{1} / t_{1}^{2}$ and $g = a (2 M + m) / m$ . After the ring catches the extra mass, the masses are equal and the motion continues at constant speed (Newton's first law): $v = h_{2} / t_{2} = a t_{1}$ . If the mass of the pulley is taken into account, $a = m g / (2 M + m + I / r^{2})$ — the “Include pulley mass” toggle gives a systematic error in the measured $g$ . The string tensions are $T_{1} = (M + m) (g - a)$ , $T_{2} = M (g + a)$ .

Bodies connected over a pulley

A block of mass $m_{1}$ on an inclined plane is connected over a pulley to a hanging mass $m_{2}$ . The force pulling on the string is $m_{2} g$ ; the opposing effects are $m_{1} g \sin θ$ and friction. If $| m_{2} g - m_{1} g \sin θ | \leq μ_{0} m_{1} g \cos θ$ , the system stays at rest; otherwise $a = (m_{2} g - m_{1} g \sin θ \mp μ m_{1} g \cos θ) / (m_{1} + m_{2})$ . If $m_{2} = m_{1} (\sin θ + μ \cos θ)$ , a block given a push moves steadily up the slope. When the hanging mass touches the floor, the string goes slack and the block is slowed only by its weight and friction.

Rolling and the moment of inertia

For a body rolling without slipping, $m g h = \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2}$ and $v = ω r$ , so $a = g \sin θ / (1 + k)$ , where $k = I / (m r^{2})$ : $k = 2 / 5$ for a sphere, $1 / 2$ for a solid cylinder, $1$ for a thin-walled tube. Whatever the mass and radius, the sphere arrives first and the tube last. Lab work: from the time $t$ taken to roll a distance $d$ we find $k = g t^{2} \sin θ / (2 d) - 1$ and $I = k m r^{2}$ (in detail — in the “Dynamics of a rigid body” experiment). Model: the string and pulley are massless (unless the toggle is on), $μ$ is constant, there is no air resistance, and the rolling bodies do not slip.

Did you know?

The Atwood machine was described in 1784 by the English mathematician George Atwood: because the difference between the masses is small, the motion is slow, and the acceleration of free fall could be measured with an ordinary clock.
Galileo studied the law of free fall using balls rolling along an inclined groove: the slope slowed the motion down and made it possible to measure that the distance is proportional to the square of the time (“Dialogues Concerning Two New Sciences”, 1638).

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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