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Velocity Ratio

Explore how machines trade input travel distance for output force or speed. Simulate gear train rotations, wheel windlasses, and third-class levers to visualize velocity ratios and trace displacement relationships.

Distance & Speed Displacement Lab ⓘ

Rotate gears, spin windlasses, or pivot speed-multiplying levers. Trace linear and angular velocity relations.

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Live Telemetry

Velocity Ratio (VR)
1.00
Effort Travel
0.0°
Load Travel
0.0°
Effort Speed
0.0°/s
Load Speed
0.0°/s
Est. Real MA
0.88
Efficiency
88 %
Machine Type
Gear Train

Introduction to Velocity Ratio

In the study of simple machines, the Velocity Ratio (VR)—also referred to as the velocity ratio of displacement—is a core geometric factor that compares the distance traveled by the effort force to the distance traveled by the load force in the same time interval.

Unlike Mechanical Advantage, the Velocity Ratio of a machine is independent of friction or bearing wear. It is a constant, ideal value determined purely by the machine\'s structural design (e.g. the diameters of pulleys, ratios of gear teeth, or leverage dimensions).

Mathematical Formulation

The basic equation for the Velocity Ratio of any machine is:

Velocity Ratio (VR) = dEffort / dLoad = vEffort / vLoad

Where:

  • dEffort is the displacement distance moved by the effort force.
  • dLoad is the displacement distance moved by the load weight.
  • vEffort and vLoad are the respective velocities.

Velocity Ratio vs. Mechanical Advantage

While Velocity Ratio (VR) and Ideal Mechanical Advantage (IMA) are equal, the Actual Mechanical Advantage (AMA = FLoad / FEffort) is always lower in practice. The correlation between the three properties is governed by the machine\'s efficiency (η):

Efficiency (η) = (AMA / VR) · 100%

Solved Numerical Examples

Example 1

A bicycle has a gear train with a 44-tooth driver chainring (connected to the pedals) and a 16-tooth driven sprocket (connected to the rear wheel). Calculate: (a) the velocity ratio of this gear system, (b) the number of rotations the rear wheel completes when the cyclist makes 5 full pedal rotations, and (c) the velocity ratio when shifted to a climbing gear with a 22-tooth chainring and 32-tooth sprocket.

View Step-by-Step Solution
  1. Given: Driver teeth Ndriver = 44 T, driven teeth Ndriven = 16 T.
  2. (a) Calculate Velocity Ratio (VR) for the high gear:
    The velocity ratio of a gear train is defined as: VR = Ndriven / Ndriver.
    VR = 16 / 44 = 4 / 11 ≈ 0.364.
    A velocity ratio of less than 1 indicates a speed/distance multiplier: the output wheel rotates faster and further than the input pedal.
  3. (b) Calculate Rear Wheel Rotations:
    Rotations of driven = Rotations of driver / VR.
    Rotations of driven = 5 rotations / (16/44) = 5 · (44 / 16) = 13.75 rotations.
    The wheel spins 13.75 times for every 5 pedal turns.
  4. (c) Find climbing gear Velocity Ratio:
    For Ndriver = 22 T, Ndriven = 32 T:
    VR = Ndriven / Ndriver = 32 / 22 = 16 / 11 ≈ 1.455.
    Here, VR > 1, meaning it acts as a force multiplier (easier to climb hills, but wheel spins slower).
  5. Result: The high-gear VR is 0.364, spinning the wheel 13.75 times. The climbing-gear VR is 1.455.
Final Answer: High VR = 0.364; Rotations = 13.75; Climbing VR = 1.455
Example 2

A differential wheel and axle windlass has an effort wheel of diameter 40 cm, and a double-radius axle drum with diameters of 16 cm and 12 cm. The rope wraps in opposite directions around the two axle drums, suspending a moveable pulley that lifts a load weight. Calculate the velocity ratio of this differential windlass.

View Step-by-Step Solution
  1. Given: Wheel diameter D = 40 cm ⇒ Radius R = 20 cm.
    Axle drum 1 diameter d1 = 16 cm ⇒ Radius r1 = 8 cm.
    Axle drum 2 diameter d2 = 12 cm ⇒ Radius r2 = 6 cm.
  2. Understand the Differential Principle:
    As the wheel rotates once, the effort rope travels 2πR. On the axle, one side winds rope in while the other unwinds it, causing the moveable pulley to lift by a net travel distance of π(r1 - r2).
    Therefore, the velocity ratio formula is:
    VR = 2R / (r1 - r2).
  3. Calculate the Velocity Ratio (VR):
    VR = 2 · 20 cm / (8 cm - 6 cm) = 40 / 2 = 20.0.
    This means for every 20 cm of effort rope pulled, the load rises by exactly 1 cm. This provides an enormous force magnification in a compact device.
  4. Result: The velocity ratio of the differential windlass is 20.0.
Final Answer: VR = 20.0
Example 3

A third-class lever (a speed multiplier) has a total length of 150 cm. The fulcrum is at one end. An effort force is applied vertically upwards at a distance of 30 cm from the fulcrum, lifting a load placed at the opposite tip. (a) Calculate the velocity ratio of this lever. (b) If the load is raised vertically by 25 cm, how far does the effort actuator move?

View Step-by-Step Solution
  1. Given: Effort arm dE = 30 cm, load arm dL = 150 cm, load lift distance hL = 25 cm.
  2. (a) Calculate Velocity Ratio (VR):
    For a lever, Velocity Ratio is the ratio of effort arm to load arm: VR = dE / dL.
    VR = 30 cm / 150 cm = 0.20.
    A velocity ratio of 0.2 indicates that the effort moves only 0.2 times the distance of the load (speed and distance are magnified by 5 times at the load tip).
  3. (b) Find Effort Travel Distance (hE):
    By definition: VR = hE / hL.
    hE = VR · hL = 0.2 · 25 cm = 5.0 cm.
    The effort actuator moves only 5.0 cm to sweep the load tip through a 25 cm arc.
  4. Result: The lever's velocity ratio is 0.20, requiring an effort travel of 5.0 cm.
Final Answer: VR = 0.20; hE = 5.0 cm

Conceptual Practice

Q1

Why does the velocity ratio (VR) of a machine remain constant while its mechanical advantage (MA) varies?

Show Explanation

The Velocity Ratio (VR) is determined solely by the fixed geometric dimensions of the machine (such as gear tooth counts, wheel diameters, or lever arm lengths). Since these physical components do not change shape or size during operation, the VR is constant. Mechanical Advantage (MA), however, depends on friction, wear, speed, and load weight, which fluctuate during use and alter energy losses.

Q2

Under what condition will the Velocity Ratio of a machine be exactly equal to its Actual Mechanical Advantage?

Show Explanation

The Velocity Ratio (VR) equals the Actual Mechanical Advantage (AMA) only in an ideal, frictionless machine with zero energy losses. In such a hypothetical system, the mechanical efficiency is 100%, meaning all input work is converted to output work (Work Input = Work Output ⇒ F_effort · d_effort = F_load · d_load ⇒ F_load / F_effort = d_effort / d_load ⇒ AMA = VR).

Q3

What does a Velocity Ratio of less than 1 (VR < 1) represent? Give a real-world example.

Show Explanation

A Velocity Ratio of less than 1 (VR < 1) means the distance moved by the effort is less than the distance moved by the load. This indicates that the machine acts as a speed and distance multiplier rather than a force multiplier. Examples include a third-class lever (like tweezers, tongs, or the human forearm) and a bicycle shifted to high gear.

Q4

If a machine has a Velocity Ratio of 6.0 and an efficiency of 75%, what is its Actual Mechanical Advantage?

Show Explanation

We use the efficiency formula: Efficiency (η) = (AMA / VR) · 100%.
Rearranging the formula to solve for AMA gives:
AMA = VR · (η / 100) = 6.0 · (75 / 100) = 6.0 · 0.75 = 4.5.
The machine multiplies the input force by 4.5 times instead of the theoretical 6.0 due to friction.

Q5

How is the Velocity Ratio of a multi-stage pulley system determined?

Show Explanation

The Velocity Ratio of a block and tackle pulley system is equal to the number of supporting rope segments directly supporting the moveable pulley blocks. For example, a system with 4 supporting rope segments has a VR of 4.0, meaning the operator must pull 4 meters of rope to raise the load by 1 meter.

Frequently Asked Questions

What is Velocity Ratio?

Velocity Ratio (VR) is the ratio of the distance moved by the effort force to the distance moved by the load force in a machine.

What is the formula for Velocity Ratio?

The formula is: Velocity Ratio (VR) = Distance moved by Effort / Distance moved by Load.

Does friction affect the Velocity Ratio?

No. Velocity Ratio is a purely geometric property based on the physical dimensions of the machine. Friction only affects the output force (AMA) and efficiency, not the travel distance ratio.

What does a VR > 1 mean?

A VR greater than 1 means the effort moves further than the load. The machine acts as a force multiplier, making it easier to lift heavy loads with less effort force.

What does a VR < 1 mean?

A VR less than 1 means the effort moves a shorter distance than the load. The machine acts as a speed and distance multiplier, magnifying movement at the expense of requiring more effort force.

How is VR related to Mechanical Advantage?

Velocity Ratio represents the ideal force multiplication under zero friction. They are related via efficiency: Actual Mechanical Advantage (AMA) = Velocity Ratio (VR) * Efficiency.

What is the VR of a gear train?

The VR of a gear train is the ratio of the number of teeth on the driven gear to the number of teeth on the driver gear: VR = N_driven / N_driver.

LeverPulleyInclined PlaneWheel & AxleMechanical AdvantageEfficiency of Machine