Balances with friction: equilibrium on equal and unequal arms

Balancing weights on equal arms with friction

Under 'On simple equality', a and g weigh 4 pounds (2 of the roller and 2 of the hung weight), giving one pound of friction marked at b. Leonardo shows that although the equal arms carry 3 pounds against 2, the 3 never wins because a further quarter (3/4) goes into friction, so an added pound is needed at e whose quarter, joined to 3/4, restores the balance.

Centre of gravity of a body on a single cord

Beside a figure, Leonardo states that the centre of gravity of any heavy body suspended by a single cord falls below the central line of that cord. He repeats that the centre of every suspended heavy body falls below the centre of the single cord that sustains it.

Working the friction fractions

In a case marked 'by 7', Leonardo divides 3/8 by 7 to get 3/56, takes this as the quarter of the friction, and clothing it obtains 12/56 for the added weight below. Adding the 3/56 to the 21/56 already found yields 24/56 of friction against 12/56 of weight.

Scaling friction with the length of the balance arm

Leonardo gives a general rule: because on an equal arm the 3/4 weight is divided into 3 parts, on a double arm the friction is divided into 7, on a triple into 11, quadruple 15 and quintuple 19. One of the fractions is always to be 'clothed' with its body by multiplying by 4, since a sub-quadruple part of the body goes into friction on the roller.

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Nel margine destro in alto tre figure con lettere e numeri.

Al centro figura con lettere e numeri:

A lato della figura:

Nel margine destro, al centro del foglio, figura con numeri.

A lato della figura:

In basso a destra quattro figure con numeri.

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Figura con lettere e numeri:

A lato della figura:

Figura con numeri:

A lato della figura:

Figura con numeri.

A lato della figura:

Sotto la figura:

In the right margin, at the top, three figures with letters and numbers.

At the centre, a figure with letters and numbers:

Beside the figure:

In the right margin, at the centre of the sheet, a figure with numbers.

Beside the figure:

At the bottom right, four figures with numbers.

Figure with letters and numbers:

Beside the figure:

Figure with numbers:

Beside the figure:

Figure with numbers.

Beside the figure:

Below the figure:

Balances with friction: equilibrium on equal and unequal arms

On this page

Balancing weights on equal arms with friction

Centre of gravity of a body on a single cord

Working the friction fractions

Scaling friction with the length of the balance arm