law of conservation of energy derivation
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Hi there,
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time. This law means that energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another.
hope i am right
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time. This law means that energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another.
hope i am right
ayushi1275:
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time. This law means that energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another.
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It is a direct consequence of Newton’s Second Law:
(1) F=ma=md2xdt2F=ma=md2xdt2
where I am using boldface to represent vector quantities, and x=x(t)x=x(t) is the vector of time-dependent coordinates. In non-relativistic mechanics, the energy is defined as
E=12mv2+V(x)=12mdxdt⋅dxdt+V(x)E=12mv2+V(x)=12mdxdt⋅dxdt+V(x)
where V(x)V(x) is the potential energy. The force is the (negative) spatial derivative of the potential energy,
(2) F=−∂∂xV(x)=−∇V(x)F=−∂∂xV(x)=−∇V(x)
If we take the total time derivative of the energy,
dEdt=12mddtdx(t)dt⋅dx(t)dt+ddtV(x(t))dEdt=12mddtdx(t)dt⋅dx(t)dt+ddtV(x(t))
and apply the product rule and the chain rule,
dEdt=mdxdt⋅d2xdt2+∂∂xV(x(t))⋅dxdtdEdt=mdxdt⋅d2xdt2+∂∂xV(x(t))⋅dxdt
But, using (1) and (2), we find
dEdt=dxdt⋅F−F⋅dxdt=0dEdt=dxdt⋅F−F⋅dxdt=0
This says that the rate of change of the energy is always zero. Hence, energy is a conserved quantity.
(1) F=ma=md2xdt2F=ma=md2xdt2
where I am using boldface to represent vector quantities, and x=x(t)x=x(t) is the vector of time-dependent coordinates. In non-relativistic mechanics, the energy is defined as
E=12mv2+V(x)=12mdxdt⋅dxdt+V(x)E=12mv2+V(x)=12mdxdt⋅dxdt+V(x)
where V(x)V(x) is the potential energy. The force is the (negative) spatial derivative of the potential energy,
(2) F=−∂∂xV(x)=−∇V(x)F=−∂∂xV(x)=−∇V(x)
If we take the total time derivative of the energy,
dEdt=12mddtdx(t)dt⋅dx(t)dt+ddtV(x(t))dEdt=12mddtdx(t)dt⋅dx(t)dt+ddtV(x(t))
and apply the product rule and the chain rule,
dEdt=mdxdt⋅d2xdt2+∂∂xV(x(t))⋅dxdtdEdt=mdxdt⋅d2xdt2+∂∂xV(x(t))⋅dxdt
But, using (1) and (2), we find
dEdt=dxdt⋅F−F⋅dxdt=0dEdt=dxdt⋅F−F⋅dxdt=0
This says that the rate of change of the energy is always zero. Hence, energy is a conserved quantity.
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