Math  /  Algebra

Question(II) A lever such as that shown in Fig. 7-20 can be used to lift objects we might not otherwise be able to lift. Show that the ratio of output force, FOF_{\mathrm{O}}, to input force, FIF_{\mathrm{I}}, is related to the lengths I\ell_{\mathrm{I}} and O\ell_{\mathrm{O}} from the pivot by FO/FI=I/OF_{\mathrm{O}} / F_{\mathrm{I}}=\ell_{\mathrm{I}} / \ell_{\mathrm{O}}. Ignore friction and the mass of the lever, and assume the work output equals the work input. (a)
FIGURE 7-20 A lever. Problem 11. (b)

Studdy Solution

STEP 1

1. The lever is an ideal lever, meaning there is no friction and the mass of the lever is negligible.
2. The work output equals the work input.
3. The lever is in static equilibrium, meaning the torques are balanced.
4. We are given the lengths from the pivot to the input and output forces, I\ell_{\mathrm{I}} and O\ell_{\mathrm{O}}.
5. We need to show that the ratio of output force to input force is equal to the ratio of the lengths from the pivot.

STEP 2

1. Define the concept of torque in the context of a lever.
2. Apply the condition for static equilibrium.
3. Relate the torques to the forces and distances.
4. Solve for the ratio of forces.

STEP 3

Define the concept of torque in the context of a lever.
Torque (τ\tau) is defined as the product of force and the perpendicular distance from the pivot point. Mathematically, τ=F×\tau = F \times \ell, where FF is the force applied and \ell is the distance from the pivot.

STEP 4

Apply the condition for static equilibrium.
For a lever in static equilibrium, the sum of the torques around the pivot must be zero. This means that the torque due to the input force must equal the torque due to the output force.

STEP 5

Relate the torques to the forces and distances.
The torque due to the input force is τI=FI×I\tau_{\mathrm{I}} = F_{\mathrm{I}} \times \ell_{\mathrm{I}}.
The torque due to the output force is τO=FO×O\tau_{\mathrm{O}} = F_{\mathrm{O}} \times \ell_{\mathrm{O}}.
Since the lever is in static equilibrium, τI=τO\tau_{\mathrm{I}} = \tau_{\mathrm{O}}.

STEP 6

Solve for the ratio of forces.
Set the torques equal to each other:
FI×I=FO×O F_{\mathrm{I}} \times \ell_{\mathrm{I}} = F_{\mathrm{O}} \times \ell_{\mathrm{O}}
Rearrange to find the ratio of forces:
FOFI=IO \frac{F_{\mathrm{O}}}{F_{\mathrm{I}}} = \frac{\ell_{\mathrm{I}}}{\ell_{\mathrm{O}}}
This shows that the ratio of the output force to the input force is equal to the ratio of the lengths from the pivot.

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