A block attached to a spring with unknown spring constant oscillates with a period of 2.0 s. What is the period if The spring constant is doubled? Parts a to d are independent questions, each referring to the initial situation.

검증된 단계별 안내

Step 1: Recall the formula for the period of a mass-spring system:

T = 2 π \sqrt \frac{m}{k}

, where

T

is the period,

m

is the mass, and

k

is the spring constant.

Step 2: Understand the relationship between the period and the spring constant. The period is inversely proportional to the square root of the spring constant, as shown in the formula:

T \propto \frac{1}{\sqrt}

Step 3: If the spring constant is doubled, the new spring constant becomes

k' = 2 k

. Substitute this into the formula for the period:

T' = 2 π \sqrt \frac{m}{k}

Step 4: Simplify the expression for the new period. Since

k' = 2 k

, the new period becomes

T' = 2 π \sqrt \frac{m}{2}

, which simplifies further to

T' = \frac{T}{\sqrt}

Step 5: Conclude that the new period is the original period divided by the square root of 2. This means the oscillation will be faster when the spring constant is doubled.

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주요 개념

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

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In SHM, the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. This motion is characterized by a constant period, which depends on the mass of the object and the spring constant.

Period of Oscillation

The period of oscillation is the time taken for one complete cycle of motion in a periodic system. For a mass-spring system undergoing SHM, the period (T) is given by the formula T = 2π√(m/k), where m is the mass attached to the spring and k is the spring constant. This relationship shows that the period is influenced by both the mass and the stiffness of the spring.

Effect of Spring Constant on Period

Doubling the spring constant (k) affects the period of oscillation in a mass-spring system. According to the formula T = 2π√(m/k), if k is increased, the period T decreases, indicating that the system oscillates faster. Specifically, if the spring constant is doubled, the new period becomes T' = 2π√(m/(2k)), which is T/√2, showing a direct relationship between the spring constant and the oscillation frequency.