# Validity of work-energy theorem in presence of non-conservative forces?

How can the work-energy theorem be valid in presence of non-conservative forces since conservation of energy is not there?

• Conservation of energy is not valid but work-energy theorem is always valid here. – user36790 Nov 29 '14 at 8:37
• Work-energy theorem tells how much work is done by the net force on the body. So, it is still applicable for friction. – user36790 Nov 29 '14 at 8:44
• "Conservation of energy is not valid"-Are you sure about this? Mechanical energy is not conserved, but energy is conserved, isn't it? – Immortal Player Nov 29 '14 at 10:58
• @Godparticle: Law of Conservation of energy is always valid. I am saying about the block system. Even the floor will radiate it to the surroundings! So, the system's definition must be expanded. – user36790 Nov 29 '14 at 12:35
• The work-energy theorem is not derived from the notion of conservative forces, it comes directly from Newton's 2nd law and basic kinematics. – dmckee --- ex-moderator kitten Mar 7 '15 at 22:50

How can the work energy theorem be valid in presence of non-conservative forces since conservation of energy is not there?

The work-energy principle simply states that work is the net increase of KE

The principle of work and kinetic energy (also known as the work-energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle

Imagine that the black force on the right (14 N) is the force of gravity acting on a body B, and that the red opposing force on the left (-10 N) is air drag, which is not conservative. The net resulting force (obtained by vector addition is +4N and this will determine an increase of KE. The principle just says that that increase is the work done on B.

The energy lost has been transformed into other forms of energy (heat, sound etc), and the general principle of conservation of energy is valid, but has nothing to to with the work-energy principle

The work-energy principle is valid regardless of the presence of any non conservative forces. As long as you are using the work done by the resultant force (and resultant moment when involving rigid bodies) in the equation (or equivalently adding the work done by each force/moment), the work energy principle is valid. This can be shown by considering a simple 1-dimensional example.

Consider a particle of mass $m$ that travels between points 1 and 2. It travels in a straight line, and has displacement for an origin $x$, a velocity $v$. It is acted upon by some forces, including conservation forces, $F_c$ and non-conservative forces, $F_{nc}$. Therefore, a net force $F_{net} = F_c + F_{nc}$ acts on the particle. By Newton's 2nd Law, note $F_{net} = ma$, where $a$ is acceleration.

Work is defined as the product of the force multiplied by the component of displacement in the direction of force, which is $F_{net} \cdot \Delta x$ for a constant force. If force is to vary over the path, then you must instead use $\int^{x_2}_{x_1}{F_{net} \cdot dx}$.

$$Work, W = \int^{x_2}_{x_1}{F_{net} \cdot dx} = m \int^{x_2}_{x_1}{a \cdot dx}$$

$$a = \frac{dv}{dt} = \frac{dx}{dt}\frac{dv}{dx} = v\frac{dv}{dx}$$ $$\therefore a \cdot dx = v \cdot dv$$

$$W = m \int^{v_2}_{v_1}{v \cdot dv} = m \left[ \frac{1}{2}v^2 \right]^{v_2}_{v_1}$$

$$\therefore W = \frac{1}{2}mv^2_2 - \frac{1}{2}mv^2_1 = \Delta KE$$

While this example is set for a 1-D problem, it can be expanded to 2 or 3-dimensions by setting the work as $\int^{x_2}_{x_1} \vec F_{net} \cdot \vec{dx}$ instead, using vector calculus, to obtain the same result.