Fundamental Physics by Ramamurti Shankar at Yale_2_Energy Conservation

The second part of his lecture talks about Energy starting with kinetic energy.

There are two concepts to understand – power and energy. Energy is deduced from the motion equation in last topic to be K2-K1=F*d, here, F*d is the work that is done, giving energy. it can be measured up by F/Newton and d/Meter, so one newton meter is an energy unit, we name is after the first scientist Joule.

Take the derivation on both sides, we get the rate of motion/work being done dk/dt equals F*dx/dt or F*velocity, this metrics defines the rate, later on, coined after another scientist Watt’s name – Watts. One unit of Watts equals to one Newton * Meter / Second.

To deduce the formula to calculate the kinetic energy, using the above concepts of energy and work, we have (cited from quora)

Considering in reality, the friction force in playing, the formula is

Worth noting that if we need to calculate the f(x) value such as plotted below, using derivative framework, infinitely is the Taylor polynomial equation.

It is very useful way to do approximation for example, Einstein’s mass equation is not intuitive to use but after transformation as below, it’s easier.

Next, he starts to look into the situation where the f is composed of two variables x and y. Concrete example of such cases are altitude and latitude on the earth, the temperature f is dependent on both of these two variables. now the questions is for a minor change of delta x and delta y, how to deduce the change of f?

In energy conservation, we can deduce the rate of kinetic energy deltak/deltat (power) = Fx*Vx + Fy*Vy. Delta K, energy change would be Fxdx +Fydy = delta W(work). Taking it to the differentiation world, as plotted blow, the direction of F and direction of moving distance delta r are not same, cos theta is needed in the calculation.

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