52. Two masses hanging side by side from springs have positions s_1 = 2 sin t and s_2 = sin 2t,
respectively.

a. At what times in the interval 0 < t do the masses pass each other? (Hint: sin 2t = 2 sint cost.)

Guida verificata passo dopo passo

To find the times when the masses pass each other, we need to set their positions equal: s_1 = s_2. Given s_1 = 2 sin t and s_2 = sin 2t, we equate them: 2 sin t = sin 2t.

Use the hint provided: sin 2t = 2 sin t cos t. Substitute this into the equation: 2 sin t = 2 sin t cos t.

Simplify the equation by dividing both sides by 2 sin t, assuming sin t is not zero: 1 = cos t.

Solve the equation 1 = cos t for t in the interval 0 < t. The cosine function equals 1 at t = 0, but we need to find other solutions within the interval.

Since cos t = 1 at t = 0 and t = 2π, within the interval 0 < t, the solution is t = 2π. However, check for other possible solutions within the given interval.

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Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and physics, representing periodic phenomena. In this problem, the positions of the masses are given by sine functions, which oscillate between -1 and 1. Understanding how these functions behave over time is crucial for determining when the two masses pass each other.

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Percorso guidato

6:04

Introduction to Trigonometric Functions

Equating Positions

To find when the two masses pass each other, we need to set their position equations equal to each other: s_1 = s_2. This involves solving the equation 2 sin(t) = sin(2t). This step is essential as it translates the physical scenario into a mathematical problem that can be solved using algebraic techniques.

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Solving Logarithmic Equations

Double Angle Identity

The double angle identity for sine states that sin(2t) = 2 sin(t) cos(t). This identity simplifies the equation we need to solve, allowing us to express sin(2t) in terms of sin(t) and cos(t). Utilizing this identity is a key step in solving the problem, as it helps to reduce the complexity of the equation.

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