Unizor - Physics4Teens - Mechanics - Power
Channel: Zor Shekhtman on UNIZOR Education
Date: 2018-12-26

Unizor - Physics through a Looking Glass of Mathematician

Notes to a video lecture on http://www.unizor.com

To analyze the mechanical work performed to achieve certain results, we often use a concept of power.
Let's assume that something or someone performs certain work and, as the time goes by, the work performed is a function of time W(t).
The power is the rate, at which the work is performed.
If during the time t the work performed is W, we define the average power of the whoever or whatever performs the work as
P = W/t
Most likely, at equal in length but different time intervals the amount of work performed will be different. For example, when a car starts, its engine should give a car an acceleration, which requires more work per unit of time than to maintain a constant speed on a smooth straight road.
In cases like this we can talk about an instantaneous power as a function of time that can be calculated using differentials:
P(t) = dW(t)/dt

Consider an example of an object in uniform motion against the force of friction with a constant speed V.
The force F that moves it forward must be equal in magnitude and opposite in direction to the force of friction to maintain the constant speed. Since the friction is constant, the force F must be constant as well.
The distance S covered as a function of time t is
S(t) = V·t
Therefore, the work performed by the force F during the time t is
W(t) = F·S(t) = F·V·t
From the definition of power follows that the power this force F exhorts is
P(t) = dW(t)/dt = F·V
As an example, the car engine exhorts the same power and consumes the same amount of gas per unit of time, if the car uniformly moves along a straight road. This power is used to generate a force sufficient to overcome the friction of wheels and air resistance.

Consider a more general case, when the motion is not uniform.
Assume, an object of mass m moves as a result of action of force F(t), where t is time. The distance it covers is S(t).
Then during an infinitesimal time interval from t to t+dt the work performed by this force will be
dW(t) = F(t)·dS(t)
Considering the Newton's second law,
F(t) = m·a(t),
where a(t) is acceleration.
Increment of distance is
dS(t) = V(t)·dt,
where V(t) is an object's speed.
Also, by definition of acceleration,
a(t) = dV(t)/dt
Therefore,
dW(t) = (m·dV(t)/dt)·V(t)·dt =
= m·V·dV(t)
Power exhausted by this force is, therefore
P(t) = dW(t)/dt =
= m·V·dV(t)/dt =
= m·V(t)·a(t)

From the definition of power as amount of work per unit of time or, more precisely, the first derivative of work by time
P(t) = dW(t)/dt
follows that the unit of measurement of power is joule/sec called watt.
Expanding the definition of joule as newton·meter,
1 watt = 1J/sec = 1N·m/sec
Obvious extensions of unit of power watt are
kilowatt = 1,000 watt and
megawatt = 1,000,000 watt.

There is an old unit of power called horsepower.
Metric horsepower, derived from lifting up against a force of gravity on Earth a weight of mass 75 kg with a constant speed of 1 m/sec, is related to watt unit as
1(metric HP) =
=75(kg)·9.8(m/sec²)·1(m/sec)≅
≅ 735.5(W)
For historical reasons there is also a mechanical horsepower, defined as 33,000 pound-feet per minute, related to watt unit as
1(mechanical HP) ≅ 745.7(W)

So, a car engine of 200 mechanic horsepower has the power of about 149,140 watt.

Watt, as a unit of measurement, was called in honor of James Watt, an 18th century Scottish scientist who was one of the first to research a concept of power, developed steam engines and measured the power of a horse.

Now let's address the concept of power in a case of rotation with constant angular speed. An example is lifting a bucket of water from a well.
Assume, a bucket of water has a mass m and we lift it with constant linear speed V with an angular speed of the well's wheel ω=V/R, where R - radius of a well's wheel.

Since the speed is constant, the force F that acts on a bucket equals to m·g, where g - acceleration of the free fall.
At the same time, if the wheel is turned by some motor and R is the radius of its shaft, the motor manufacturer provides technical characteristic not only of the power, but also of a torque of a motor.
Remember that the torque equals to
τ = F·R

So, on one hand, we have expressed the power of a motor P in terms of unknown force F and linear speed of a bucket:
P = F·V = F·R·ω
On another hand, we expressed the torque of this motor in terms of the same unknown force F and a radius of its shaft:
τ = F·R
Substituting torque τ for F·R in the formula for power, we can find the relationship between the power of a motor and is torque:
P = τ·ω

Notice the similarity between the formula for power in case of uniform motion along a straight line P=F·V and formula for power in case of rotation with constant angular speed P=τ·ω. Instead of force F in case of straight line motion, we use torque τ for rotation and, instead of linear speed V for straight line motion, we use angular speed ω for rotation.