Mathematics of Mechanical Power

Mathematics of Mechanical Power

Just as the mechanical energy, mechanical power is obtained from a combination of both forces and movement in a mechanical system. Mechanical power can therefore be described as the time derivative of work. The work done by a force F on an object that travels along a curve C in mechanics, is given by:

Integration

Where x defines the path C and v is the velocity along this path. If the force F in the above equation is derivable from a conservative force, then applying the gradient theorem gives: WC = UB – UA, where A is the beginning and B is the end of the path along which the work is done.

We also learnt during our Physics tuition classes, during the chapters of Gravitation and Electric field, that force is the negative of the gradient of the potential energy. Therefore the mechanical power at any point along the curve C is the time derivative and can be obtained by:

Differentiation

The above equation can be simplified as: P = F v. However, when it comes to rotational systems, mechanical power is the product of the torque and angular velocity measured in radians per second. Then in fluid systems such as hydraulic actuators, mechanical power is mathematically calculated as: P = p Q, where p is pressure in pascals, or N/m2Q is volumetric flow rate in m3/s in S.I. units.

An Example / Calculation

Consider a car engine that provides a forward force of 5000 N. If the car is moving at 40 m s-1, what power does the engine develop?

Solution: In one second, the car moves a distance of 40 m. Therefore, we can calculate the work done in one second as force × distance = 5000 N × 40 m = 200,000 J = 200 kJ. Power = work done divided by the time taken = 200 kJ / 1 s = 200 kW.

Alternatively, the equation Power = force × velocity, could also be used to obtain the power. This equation is derived from the equation: Power = work done divided by the time taken which in turn is equal to (force × displacement in direction of the force) divided by the time taken. And we know that displacement in the direction of the force / time taken = velocity. Giving the equation, power = force × velocity. Velocity = 40/1 = 40 m s-1 while Power = 5000 (force) × 40 (velocity) = 200, 000 W = 200 kW.