What is the Velocity Ratio formula of a machine?

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Velocity Ratio Formula is the topic of this physics notes. Here we will see what Velocity Ratio (V.R.) is and then quickly state its formula or equation.

V.R. is a term related to a machine that gives an idea of whether there is any speed gain or force multiplication or direction change of the effort for that machine.

V.R. lets us know if the displacement of load is equal or more or less than the displacement of effort. Now let’s find out what is the formula or equation of that ratio.

What is the Velocity Ratio formula?

The velocity ratio (V.R.) of a machine is defined as the ratio of the velocity of the effort to the velocity of the load. In other words, it is the ratio of the displacement of the effort to the displacement of the load. Velocity Ratio Formula is represented with 2 equations listed below.
1) V.R. = velocity of the effort / the velocity of the load = VE  / VL …… (1)
2) V.R. = displacement of the effort / displacement of the load = dE  / dL …….. (2)

Deriving the Velocity Ratio formula as displacement ratio

We will start with the equation of the Velocity Ratio or V.R. as the ratio of 2 velocities, as per its definition.
V.R. = velocity of the effort / velocity of the load = VE  / VL

Now, if the effort makes a displacement dE in time t, then the velocity of effort VE = dE/t
similarly, if the load makes a displacement dL in time t, then the velocity of load VL = dL/t

Hence, V.R. = VE  / VL = (dE/t) / (dL/t) =dE  / dL

This is how we can prove that V.R. is represented by another displacement ratio formula which is like this: V.R. = dE  / dL = ratio of the displacement of the effort to the displacement of the load.

Speed Gain of machines | VR value of Speed multiplier machines

VR <1
When the Velocity Ratio (VR) value of a machine is less than 1 then we can say that machine provides a speed gain.

From equation 1 above you can say that in this case, the velocity of the load is more than the velocity of the effort.

And as per equation 2 above you can also say that the displacement of load > displacement of effort.

This refers to speed gain. Think about the pair of blades of scissors as an example.

When VR<1 for ideal machines, then MA is also <1. So these machines are notforce multipliers‘. As these machines provide speed gain as said above, these are also known as speed multipliers.
Examples: all levers of class 3 ( Tong, spade used for lifting a load), Scissors with long blades (it’s a class I lever with load arm longer than its effort arm)

Force Multiplication in machines and VR value for these machines

V.R. >1
When the Velocity Ratio (V.R.) value of a machine is more than 1 then we can say that machine provides a force multiplication.

From equation 1 above you can say that in this case, the velocity of the effort is more than the velocity of the load.

And as per equation 2 above you can also say that the displacement of effort > displacement of the load.

For an ideal machine with VR >1, we can say the mechanical advantage(MA) of it is also more than 1. Obviously it indicates a force multiplication. (as load > effort for MA >1)
These machines whose MA>1 are called force multipliers.

Examples: all levers of class 2 (ex. nutcracker, wheelbarrow, bar to lift load), some class 1 levers (crowbar, pliers)

Related readings:

Types of levers

MA of levers


What is the Velocity Ratio formula of a machine?
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