Questions: Conservation of Mechanical Energy in Systems

Question 1

A bead slides without friction along a curved wire from rest at a height of 2 m down to ground level. What is its speed at the bottom, given g = 9.8 m/s²?

• A. It depends on the wire's shape — a steeper path accelerates the bead more and produces higher speed
• B. v = √(2gh) ≈ 6.3 m/s, determined entirely by the height difference via energy conservation
• C. v = √(gh) ≈ 4.4 m/s, since kinetic energy equals mgh/2
• D. The speed cannot be determined without knowing the normal force the wire exerts

Answer: (B)

Energy conservation: mgh = ½mv², so v = √(2gh) = √(2·9.8·2) ≈ 6.3 m/s. The wire's shape is irrelevant because the normal force is always perpendicular to the bead's motion — it does no work and never appears in the energy equation. This is the power of energy methods: constraint forces vanish entirely. Option D represents the Newtonian burden — you'd need the normal force to apply F = ma, but energy conservation sidesteps it.

Question 2

Why does the conservation of mechanical energy fail when a block slides down a ramp with significant friction?

• A. Friction is a constraint force that doesn't do work, so it shouldn't affect energy conservation
• B. Friction is a non-conservative force that converts mechanical energy to heat, so T + V decreases rather than remaining constant
• C. Friction only affects the direction of motion, not the speed, so energy is conserved but momentum is not
• D. Friction changes the block's mass, violating the assumption that m is constant

Answer: (B)

Conservation of mechanical energy (T + V = constant) holds only when all forces doing work are conservative — meaning their work depends only on position, not path. Friction is non-conservative: it always opposes motion, does negative work equal to friction force times distance traveled, and converts mechanical energy into heat. The correct equation becomes T₁ + V₁ + W_friction = T₂ + V₂, where W_friction is negative. Option A is wrong: friction acts parallel to motion (along the surface), so it does do work — unlike normal forces, which act perpendicular.

Question 3

Conservation of mechanical energy (T + V = constant) applies to a pendulum because the string tension does work on the bob as it swings.

• T. True
• F. False

Answer: False

The string tension acts centripetally — always directed toward the pivot, perpendicular to the bob's velocity (which is tangential). Since force and velocity are perpendicular at every instant, the tension does zero work. Therefore the only force doing work is gravity (conservative), and T + V is conserved. The statement has the causal logic inverted: energy is conserved precisely because the tension does NO work — not because it does work.

Question 4

The energy conservation approach is more powerful than Newton's second law for constrained mechanical systems because it produces a scalar equation that never requires computing constraint forces.

• T. True
• F. False

Answer: True

Newton's second law gives vector equations — often three coupled differential equations per body — and requires identifying all forces, including constraint forces (normal forces, string tensions) that do no work. Energy conservation produces a single scalar equation T₁ + V₁ = T₂ + V₂ in which constraint forces never appear. For a bead on a wire or a ball on a ramp, the Newtonian approach carries these irrelevant forces through the calculation; energy methods skip them entirely. This is why energy methods are said to be more powerful for constrained systems.

Question 5

Explain why conservation of mechanical energy is described as a scalar conservation law, and why this matters for solving mechanics problems.

Model answer: Energy is a scalar — it has magnitude but no direction. Kinetic energy T = ½mv² and potential energy V = mgh are both scalars, so T + V = constant is a single algebraic equation with no directional components. Newton's second law F = ma is a vector equation giving one equation per spatial dimension — three coupled differential equations in 3D. Scalar energy methods collapse multi-dimensional dynamics into a single equation, eliminating component resolution and often making the problem solvable by algebra rather than integration.

The scalar nature is both a strength and a limitation. Energy conservation tells you the speed at any configuration but not the trajectory — you don't get position as a function of time. For many engineering problems, that speed information is exactly what you need (e.g., how fast does a roller coaster reach the bottom?). For others, you need the full trajectory, requiring Newton's laws or the Lagrangian. The Lagrangian formalism, which you'll meet next, extends this scalar approach to derive the full equations of motion from energy alone.