The summation of the dot product of tension and velocity in a pulley system is equal to zero due to the principle of conservation of mechanical energy.
In an ideal pulley system (one without friction or mass), the tension in the rope or string remains constant throughout. Let's consider a simple case with two masses connected by a rope passing over a pulley.
The dot product of tension and velocity can be expressed as:
T₁ · v₁ + T₂ · v₂ = 0
Here, T₁ and T₂ represent the tensions in the rope on either side of the pulley, while v₁ and v₂ represent the velocities of the masses connected to the rope.
The principle of conservation of mechanical energy states that in an ideal system, the total mechanical energy remains constant. In this case, mechanical energy is defined as the sum of potential energy (mgh) and kinetic energy (½mv²).
When applying the principle of conservation of mechanical energy to the pulley system, any gain in potential energy of one mass is offset by an equal decrease in potential energy of the other mass. The work done by tension is responsible for transferring this potential energy between the masses. Since the net gain or loss of potential energy is zero, the work done by the tension is also zero.
When work is zero, the dot product of tension and velocity is zero as well. Mathematically, this is expressed as:
T₁ · v₁ + T₂ · v₂ = 0
Hence, the summation of the dot product of tension and velocity in a pulley system is equal to zero due to the conservation of mechanical energy.