{Use of Tech} Spring oscillations A spr​​ing hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10sin t−10cos t, where x is positive when the mass is above the equilibrium position. <IMAGE>
c. At what times is the velocity of the mass zero?

Verified step by step guidance

1. To find when the velocity of the mass is zero, we first need to determine the velocity function. The velocity is the derivative of the position function x(t) with respect to time t.
2. Given the position function x(t) = 10sin(t) - 10cos(t), we differentiate it to find the velocity function v(t). The derivative of sin(t) is cos(t), and the derivative of cos(t) is -sin(t).
3. Applying these derivatives, the velocity function v(t) is: v(t) = d/dt [10sin(t) - 10cos(t)] = 10cos(t) + 10sin(t).
4. To find when the velocity is zero, we set the velocity function equal to zero: 10cos(t) + 10sin(t) = 0.
5. Solve the equation 10cos(t) + 10sin(t) = 0 for t. This involves finding the values of t where the sum of the cosine and sine terms equals zero, which can be done by using trigonometric identities or algebraic manipulation.

Key Concepts

Velocity and Its Calculation

Velocity is the rate of change of position with respect to time. In calculus, it is found by taking the derivative of the position function. For the given function x(t) = 10sin(t) - 10cos(t), the velocity function v(t) can be determined by differentiating x(t) with respect to t, which will provide the times when the mass is momentarily at rest.

Finding Critical Points

Critical points occur where the derivative of a function is zero or undefined. In the context of the velocity function, setting v(t) = 0 will help identify the times when the mass's velocity is zero, indicating moments when the mass changes direction. This is essential for understanding the oscillatory motion of the mass attached to the spring.

Trigonometric Functions and Their Properties

The position function x(t) involves sine and cosine functions, which are periodic and have specific properties regarding their maxima, minima, and zeros. Understanding the behavior of these trigonometric functions is crucial for analyzing the motion of the mass, particularly in determining the intervals of time when the velocity is zero, as these functions oscillate between -1 and 1.