# Equations

While we don’t require you to be an expert mathematician, there are a few equations necessary to understand which muscles are invovled in an exercise and for proper exercise + volume prescription. Don’t worry, they’re not that bad.

## F = ma(cosΘ)

Force (F) is the product of mass (m), acceleration (a), and the cosine of theta, or the angle between the direction of force and direction of displacement.  The displacement is not the same as distance traveled, rather its the distance from its initial position to its final position; so while the weight’s motion during a bicep curl is technically a semicircle, we consider the displacement to be the straight vertical line between the starting and ending positions of the exercise. Force is recorded in Newtons (N), mass is measured in kilograms (kg), and acceleration, which is the change of an object's velocity over time, is measured in meters per second squared (m/s2).

I'm sure you thought you'd never have to use this stuff again after high school trigonometry, but unfortunately the angle at which the force is exerted is necessary for an accurate calculation.  The cosine of any angle is a value between 0 and 1, here are some of the most common ones:

cosine(0°) = 1 | cosine(30°) = 0.866 | cosine(45°) = 0.707 | cosine(60°) = 0.5 | cosine(90°) = 0

When the force and the motion of the exercise are in the same direction, theta is 0°.  Because the cosine(0°) = 1, it has no affect on the force calculation.  But this is not always the case and we (unfortunately) have to use some trigonometry to account for these differences.

Another thing to consider are which units we're using - always remember to check and convert units

1 kg = 2.2 lbs  |  1 m = 3.28 ft

### Example 1

How much force did a woman exert if she performed scapular retraction cable rows using 80lbs at an acceleration of 0.75 m/s^2?

### Example 2

On the next set, she increased the weight by 20lbs and slowed down her acceleration to 0.5 m/s^2.  She also moved the cables up so that the angle between the extension of cables (direction of force) and the line of displacement from the machine was 60°.  How much force did she exert during this set?

## Volume = Sets x Reps x Weight

Training volume can be used to determine an appropriate workout plan that will not exceed an athlete's muscular endurance.  It looks at overall weight moved, rather than just rep range at high or low weights.  Training volume is useful for varying workout schemes while maintaining consistency to a training program.  Not every athlete is the same - some will benefit most from high weight, low reps and some will see better results with the opposite.  It's important to try out different training structures to see what works best for you or the athlete you're training.  And now instead of just guessing and experimenting with different weight and rep ranges, you can make educated changes to workout schemes based on volumes.

### Example 1

What is an athlete's workout volume if she performs 8 reps of deadlifts over 4 sets, lifting 80lbs?

### Example 2

Calculate 3 different training schemes for an athlete who fatigues after moving 3000lbs of weight.

Impulse

# Work v Power

## Work = Force x Displacement

Work is the product of the force exerted on an object and the distance the object moves in the direction of the force, while power is the rate of doing work. Work and power are considered either "positive" or "negative" depending on the directions of force and displacement. If the force exerted by the muscle and the object's displacement are in the same direction, like during the lifting phase, then the work and power are considered to be positive. But if the force and displacement are in opposite directions, like during a controlled lowering phase, then the work and power are considered to be negative.

Although it should be stated that there is actually no such thing as negative work or power.  It's just a way to refer to the work or power done on a muscle by the weight, like when the weight is going down during a bicep curl. When the weight is going up during a curl, the work is done on the weight by the muscle and it is considered positive work or power. So while you're lifting and lowering a weight, your muscle and the weight are taking turns performing work on each other. Its not the muscle performing positive and negative work. We're just calculating everything relative to the muscle by saying that work or power done on it is negative.

## Work = Force x Displacement

A man is performing a 1RM on a 45° hip sled, moving it up the track 2.5 feet.  If his 1RM is 170lbs and he's able to push the weight with an acceleration of 0.1 m/s^2, how much work is he doing?

## Power = Work/Time = Force x Velocity

For positive work and power, during the lifting phase, the total force is the sum of the force needed to overcome the weight of the bar (F1) plus the force needed to lift the bar at a given acceleration (F2).  The total force here is exerted by the muscles.  Theta in this equation is 0° because the person is lifting the bar and applying the force straight upwards.

For negative power, during the controlled lowering phase, the total force is now the combination of the downward force exerted by the bar due to gravity (F1) and the upward, resistant force exerted by the muscles to lower the bar at a given rate and prevent free-falling (F3).  The angle theta for F3 in this calculation is 180°, whose cosine turns out to be -1.  Theta for F1 is 0°because gravity is pulling the bar down, in the direction of it's motion. The typical acceleration of a deadlift is 1 foot/second, which needs to be converted to standard units to be used in the equations.

Calculate positive and negative power separately for the following example.

### Example

How much power does a person produce while doing a 220lb deadlift at a rate of 1 foot/second^2 if the bar travels 1.5m during the lifting phase, given it takes 40 seconds to complete 10 reps.

## Q = HR x SV

Cardiac output (Q) is the product of heart rate (HR) and stroke volume (SV).  Heart rate is the number of heart beats per minute and is measured with the unit bpm.  Age-predicted max heart rate is calculated by subtracting the person's age from 220.  Stroke volume is the amount of blood that the heart pumps out with every beat.  Stroke volume can be  measured using either mL/beat or L/beat, but cardiac output is always recorded in L/min.

### Example

A 30 year old man is exercising on a rowing machine at 80% of his age-predicted maximum heart rate.  What is his cardiac output if his stroke volume is 120 mL/beat?