7.2 Pulleys, Wheels and Axles, and Inclined Planes

7.2 Pulleys, Wheels and Axles, and Inclined Planes

Pulleys, inclined planes, and wheel-and-axle systems represent primary engineering solutions for lifting heavy loads vertically against gravity. In each mechanism, the mechanical advantage is achieved by trading displacement distance for effort force. Mastering these simple machines requires understanding the geometry of force vectors and the distribution of mechanical tension.

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Pulleys: Fixed, Movable, and Block & Tackle Systems

A pulley is a grooved wheel (sheave) that turns around an axle mounted inside a housing or frame (block), designed to support and guide a flexible cord, cable, or rope.

1. Single Fixed Pulley

A fixed pulley has its axle firmly anchored to an immovable overhead structure (e.g., a ceiling beam or crane mast).

• Mechanical Behavior: When pulling the rope downward, the load rises upward by the identical distance ($d_e = d_L$).
• Mechanical Advantage: $\text{IMA} = 1.0$.
• Function: It does not multiply force. Its sole advantage is changing the direction of the applied force, allowing an operator to pull downward—working with gravity and utilizing their own body weight—rather than lifting upward against gravity.

2. Single Movable Pulley

A movable pulley is attached directly to the load itself, moving upward and downward along with the load.

• Mechanical Behavior: The rope is anchored overhead at one end, loops under the movable pulley, and the free end is pulled upward by the operator.
• Tension Distribution: The total downward weight of the load ($F_L$) is shared equally between the two upward-supporting rope segments on either side of the pulley:

$T = \frac{F_L}{2} \implies F_e = \frac{F_L}{2}$

• Displacement: To lift the load by a vertical height of $h$, both supporting rope strands must shorten by $h$. Consequently, the operator must pull $2h$ of rope ($d_e = 2 d_L$).
• Mechanical Advantage: $\text{IMA} = 2.0$.
• Function: It cuts the required lifting force in half, at the expense of doubling the length of rope that must be hauled.

Single Fixed vs. Single Movable Pulley:

      FIXED PULLEY (IMA = 1)                 MOVABLE PULLEY (IMA = 2)
         [ Ceiling Anchor ]                     [ Ceiling Anchor ]
                 |                                      |
               ( O ) <--- Fixed Axle                    |      ^ Effort (Fe = W/2)
              /     \                                   |      |
             /       v Effort (Fe = W)                  \     /
            |                                            \   /
          [Load] (Weight W)                               ( O ) <--- Movable Axle
                                                            |
                                                          [Load] (Weight W)

3. Block and Tackle Systems and the Strand-Counting Rule

A block and tackle combines multiple fixed and movable pulleys into two blocks to achieve high mechanical advantage.

To determine the Ideal Mechanical Advantage of any block and tackle system on an exam, use the Strand-Counting Rule:

1. Identify the movable pulley block (the block that moves with the load).
2. Count only the rope strands that directly emerge from and support the movable block.
3. The Direction Rule: Inspect the free end of the rope where the effort is applied:
   • If the effort rope pulls downward (running off an overhead fixed pulley), it does NOT support the movable block. Do not count it.
   • If the effort rope pulls upward (running directly off a movable pulley), it DOES help support the load. Count it.

$\text{IMA} = n \quad (\text{where } n \text{ is the number of supporting rope strands})$

$\text{Ideal Effort } F_e = \frac{F_{\text{load}}}{n}, \qquad d_e = n \times d_L$

In the four-strand case, the downward pull on the free end comes off a fixed pulley, so it does not hold up the moving block. In the five-strand case, the free end pulls upward directly on the moving block, so it counts.

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Inclined Planes (Ramps)

An inclined plane is a flat, stationary supporting surface tilted at an angle $\theta$ to the horizontal. It enables a heavy load to be raised to an elevation $h$ by pushing or pulling it along the sloping ramp length $L$.

Force Vector Resolution

Gravity pulls an object of mass $m$ straight down with weight $W = mg$. On an incline, this gravitational vector is resolved into two perpendicular components:

1. Parallel Component ($F_\parallel$): Pulls the object down along the ramp surface: $F_\parallel = W \sin \theta = mg \left(\frac{h}{L}\right)$
2. Perpendicular Normal Component ($F_\perp$): Presses the object firmly into the ramp surface: $F_\perp = W \cos \theta$

Inclined Plane Force Vectors:

         /|  
        / |  
       /  |  
    L /   | h (Height)
     /    |  
    /     |  
   /___θ__|  
   Base (b)

   Gravity (W = mg) splits into:
   - Down the ramp: F_parallel = mg * sin(θ) = mg * (h / L)
   - Into the ramp: F_normal   = mg * cos(θ)

Mechanical Advantage of a Ramp

In an ideal frictionless scenario, the effort force $F_e$ needed to push the load up the ramp at constant speed must balance $F_\parallel$:

$F_e = W \left(\frac{h}{L}\right) \implies \text{IMA} = \frac{F_{\text{load}}}{F_{\text{effort}}} = \frac{L}{h} = \frac{1}{\sin \theta}$

• A ramp $10\text{ m}$ long that rises $2\text{ m}$ vertically has an $\text{IMA} = 10 / 2 = 5.0$. An operator needs only $200\text{ N}$ of ideal effort to lift a $1,000\text{ N}$ load.
• If friction is present with coefficient $\mu$, the actual effort required must overcome both gravity and friction: $F_{e,\text{actual}} = W \sin \theta + f_{\text{friction}} = W \sin \theta + \mu W \cos \theta$

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Wedges and Screws: Mobile and Helical Inclined Planes

Both the wedge and the screw are direct mechanical adaptations of the inclined plane.

1. The Wedge

While an inclined plane remains stationary while the load moves up its slope, a wedge is a mobile double inclined plane that is driven into or under a load to separate, split, or lift it.

$\text{IMA}_{\text{wedge}} = \frac{\text{Length of Wedge } (L)}{\text{Thickness / Width of Wedge } (t)}$

• A long, thin wedge has a high mechanical advantage and penetrates easily, but must be driven deep into the material to produce a given separation.
• A short, blunt wedge has a low mechanical advantage, requiring a massive driving force (e.g., from a sledgehammer), but yields rapid lateral expansion.
• Common examples include axes, chisels, knife blades, nails, doorstops, and plowshares.

2. The Screw

A screw is an inclined plane wrapped helically around a central cylinder. The spiral ridge formed by the incline is the thread.

• Pitch ($p$): The linear distance between two consecutive thread crests, representing the axial distance the screw advances into a material in exactly one full $360^\circ$ rotation.
• Mechanical Advantage: When a screw is rotated by an effort applied at the tip of a handle or lever of radius $r$ (such as a screwdriver handle or screw jack lever), the effort moves through the circumference of a circle ($2\pi r$) while the load advances by one pitch ($p$):

$\text{IMA}_{\text{screw}} = \frac{2\pi r}{p}$

Because the circumference $2\pi r$ is enormously larger than the tiny thread pitch $p$ (often $1\text{ to }5\text{ mm}$), screws generate exceptionally large mechanical advantage (often exceeding $500$ to $2,000$). Furthermore, the extensive friction between the threads prevents screws from "back-driving" under load, making them inherently self-locking (e.g., car jacks, clamps, vises, threaded bolts).

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Wheels and Axles

A wheel and axle consists of two rigidly connected concentric cylinders of different diameters that rotate together about a shared central axis.

Wheel and Axle Mechanism:

       (====== Wheel (Radius R) =====)
               |               |
               |  [= Axle =]   |
               |  (Radius r)   |
               +-------+-------+
                       |
                       v Load (Suspended on axle)

1. Applied to the Wheel (Force Multiplier)

When the input effort is applied to the perimeter of the larger cylinder (the wheel) to rotate the smaller cylinder (the axle), the system acts as a continuous rotating Class 1 or Class 2 lever:

$\text{IMA} = \frac{\text{Radius of Wheel } (R_{\text{wheel}})}{\text{Radius of Axle } (r_{\text{axle}})} = \frac{D_{\text{wheel}}}{d_{\text{axle}}}$

Because $R_{\text{wheel}} > r_{\text{axle}}$, the $\text{IMA}$ is strictly greater than 1. Effort force is multiplied at the axle. Examples include automobile steering wheels, doorknobs, screwdriver handles driving a narrow blade tip, winches, and windlasses.

2. Applied to the Axle (Speed Multiplier)

When input torque is supplied directly to the axle to turn the wheel (such as an engine driveshaft turning a car wheel, or a bicycle chain turning the rear hub), the $\text{IMA}$ is less than 1. This configuration trades force multiplication for substantial increases in linear speed and distance traveled along the wheel's outer rim.

Summary table: simple machines