Question

A body of mass \(m\) moves in one dimension under the influence of a force, \(F(x)\), which depends only on the body's position. a) Prove that Newton's Second Law and the law of conservation of energy for this body are exactly equivalent. b) Explain, then, why the law of conservation of energy is considered to be of greater significance than Newton's Second Law.

Short answer

#Answer# a) The equivalence of Newton's Second Law and the law of conservation of energy has been proven through the following steps: 1. Newton's Second Law: F(x) = m(d²x/dt²) 2. Conservation of energy: dT/dt + dU/dt = 0 3. By writing kinetic energy (T) and force (F) in terms of their derivation with respect to time, the conservation of energy equation becomes: m(d/dt(dx/dt))(dx/dt) + F(x)(dx/dt) = 0 4. Applying the chain rule and simplifying, we have: m(d²x/dt²)(dx/dt) + F(x)(dx/dt) = 0. Dividing by dx/dt, this becomes: m(d²x/dt²) + F(x) = 0, which is exactly Newton's Second Law. Thus, both laws are equivalent in this one-dimensional situation. b) The law of conservation of energy is considered more significant than Newton's Second Law because it is a more general principle that applies to various forms of energy, such as thermal, chemical, and electromagnetic energy. Furthermore, the conservation of energy law allows for easier analysis of complex systems, as it does not require resolving forces into components.

Step-by-step solution

Analyze Newton's Second Law

Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, this is written as: $$F(x) = m \frac{d^2x}{dt^2}$$ where \(x\) is the position of the object, \(t\) is time, and \(\frac{d^2x}{dt^{2}}\) is the second derivative of position with respect to time (acceleration).

Analyze the law of conservation of energy

The law of conservation of energy states that the total energy of a closed system remains constant. For a mechanical system with potential energy \(U(x)\), kinetic energy \(T\), and force \(F(x)\), the principle of conservation of mechanical energy can be stated as: $$T + U = \text{constant}$$ Taking the derivative of this equation with respect to time, we get: $$\frac{dT}{dt} + \frac{dU}{dt} = 0$$ Since kinetic energy is given by \(T = \frac{1}{2} m \left( \frac{dx}{dt} \right)^2\) and potential energy has a derivative \(-\frac{dU(x)}{dx} = F(x)\), then we can rewrite the above equation as: $$m\frac{d}{dt}\left(\frac{dx}{dt}\right)\frac{dx}{dt} + F(x)\frac{dx}{dt} = 0$$

Prove the equivalence using the chain rule

Now we will prove the equivalence of Newton's Second Law and the conservation of mechanical energy law using the chain rule on the above equation. Applying the chain rule and simplifying, we get: $$m \frac{d^2x}{dt^2} \frac{dx}{dt} + F(x)\frac{dx}{dt} = 0$$ Since \(\frac{dx}{dt}\) is the velocity of the object and it is nonzero, we can divide the entire equation by \(\frac{dx}{dt}\), which results in: $$m \frac{d^2x}{dt^2} + F(x) = 0$$ This equation is exactly the same as Newton's Second Law, which we have already derived. Therefore, Newton's Second Law and the conservation of mechanical energy law are equivalent for this one-dimensional situation.

Discuss the significance of the conservation of energy

Now that we have proven the equivalence of Newton's Second Law and the conservation of energy, let's discuss why the conservation of energy is considered to be of greater importance. The law of conservation of energy is a more general principle that applies not only to mechanical systems but also to other forms of energy, such as thermal, chemical, and electromagnetic energy. Newton's Second Law, on the other hand, specifically deals with the motion of objects due to the force applied to them. Moreover, the conservation of energy law allows for easier analysis of certain complex systems, as it does not require resolving forces into components. It is for these reasons that the conservation of energy is considered to be more significant than Newton's Second Law.

Key concepts

Newton's Second Law

Understanding Newton's Second Law of motion is crucial for solving many problems in physics. The law simply states that the force exerted on an object is equal to the mass of that object multiplied by its acceleration, often encapsulated in the equation:
\[ F = ma \] where F is the force applied, m is the mass of the object, and a is the object's acceleration. This acceleration is the rate of change of velocity, or in calculus terms, the derivative of velocity with respect to time.

When we apply this law to the movement of a body in a one-dimensional space where the force depends on the object's position, x, we can relate the force to the second derivative of that position with respect to time, which gives us a deeper look into the object's dynamics:
\[ F(x) = m \frac{d^2x}{dt^2} \]
This expression is fundamental when examining how an object’s velocity changes over time under various forces.

Mechanical Energy

The concept of mechanical energy is an essential piece of the puzzle in understanding physical systems. Mechanical energy is the sum of potential and kinetic energy in a system. Kinetic energy T is the energy of motion, calculated by the formula:
\[ T = \frac{1}{2} m \left( \frac{dx}{dt} \right)^2 \]
while potential energy U(x) is the energy stored by an object's position in a field, such as a gravitational field.

The law of conservation of mechanical energy asserts that, in the absence of non-conservative forces like friction, the total mechanical energy is conserved. This guides us to an understanding that an object’s movement and position—which determine kinetic and potential energy respectively—can transform into each other but the overall energy of the system remains constant. This principle is particularly powerful as it applies across a broad range of physical situations, not just isolated instances.

Chain Rule in Calculus

The chain rule is an invaluable tool in calculus, especially when dealing with the dynamics of physical systems. It is a method for computing the derivative of the composition of two or more functions.

For example, if a function h(t) can be written as another function g(x(t)), where x(t) itself is a function of t, then the chain rule tells us how to find the derivative of h with respect to t:
\[ \frac{dh}{dt} = \frac{dg}{dx} \cdot \frac{dx}{dt} \]
In the context of physics, and specifically for our exercise, the chain rule bridges the gap between the force exerted on an object and its energy. It allows us to differentiate complex expressions and show the equivalence between Newton's Second Law and the conservation of mechanical energy. Thanks to the chain rule, the intertwined relationships of position, force, and time can be unravelled, providing a clear mathematical path from one concept to another.