A stretched spring stores 2.0 J of energy. How much energy will be stored if the spring is stretched three times as far?

검증된 단계별 안내

Understand that the energy stored in a spring is given by the formula:

E_{s} = \frac{1}{2} k x^{2}

, where

k

is the spring constant and

x

is the displacement from the equilibrium position.

Recognize that the energy stored in the spring is proportional to the square of the displacement, as indicated by the

x^{2}

term in the formula.

If the spring is stretched three times as far, the new displacement becomes

x' = 3 x

. Substitute this into the energy formula:

E_{s}' = \frac{1}{2} k {(3x)}^{2}

Simplify the expression for the new energy:

E_{s}' = \frac{1}{2} k (9 x^{2}) = 9 \frac{1}{2} k x^{2}

Compare the new energy to the original energy. Since the original energy is

E_{s} = \frac{1}{2} k x^{2}

, the new energy is 9 times the original energy. Multiply the original energy (2.0 J) by 9 to find the new energy.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to the distance it is stretched or compressed from its equilibrium position, represented mathematically as F = kx, where F is the force, k is the spring constant, and x is the displacement. This principle is fundamental in understanding how springs behave under different loads.

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring when it is stretched or compressed. The formula for elastic potential energy (U) is U = 1/2 kx², where k is the spring constant and x is the displacement from the equilibrium position. This concept is crucial for calculating the energy stored in a spring based on its deformation.

Energy Scaling with Displacement

The energy stored in a spring scales with the square of the displacement. This means that if the displacement is tripled, the energy stored increases by a factor of nine (since (3x)² = 9x²). Understanding this relationship is essential for predicting how much energy will be stored when the spring is stretched to different lengths.