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Springs are fascinating mechanical devices that can store and release energy through their motion. When a weight is attached to a spring and is set in motion, the spring stretches and compresses, creating a back-and-forth motion. This is a type of harmonic motion, and it can be described mathematically using a displacement function. The function provided in the exercise is \[x = 10 \cos t,\]where \(x\) is the displacement measured in centimeters, and \(t\) is the time in seconds.

To find out how far the spring has moved from its equilibrium point at different times, substitute the given time values into the displacement equation. The cosine function, \(\cos t\), tells us how much the spring is displaced from the neutral position, multiplying it by 10 gives the displacement in centimeters. For instance:

• At \(t = 0\), \(x = 10 \times \cos(0) = 10\) cm. The spring is stretched to its maximum position.
• At \(t = \pi/3\), \(x = 10 \times \cos(\pi/3) = 5\) cm. The displacement is half the amplitude.
• At \(t = 3\pi/4\), \(x = 10 \times \cos(3\pi/4) \approx -7.07\) cm. The negative sign indicates the spring is on the opposite side of the equilibrium position.

This illustrates how the spring oscillates over time, with a maximum displacement of 10 cm.

Velocity is crucial in understanding how fast an object is moving and in which direction. In the context of harmonic motion for a spring, velocity comes from the derivative of the displacement function. Using the given formula \( x = 10 \cos t \), the velocity \( v(t) \) can be derived as \[v(t) = \frac{dx}{dt} = -10 \sin t.\]This shows that velocity is proportional to the sine of time and indicates how the speed of the spring's movement varies.

With this formula, you can calculate the velocity at specific times. For example:

• At \(t = 0\), \(v = -10 \sin(0) = 0\) cm/s. The spring is momentarily stationary as it changes direction.
• At \(t = \pi/3\), \(v = -10 \times \sin(\pi/3) \approx -8.66\) cm/s. This suggests the spring is moving fast in the opposite direction.
• At \(t = 3\pi/4\), \(v = -10 \times \sin(3\pi/4) \approx -7.07\) cm/s. The spring continues its motion, maintaining significant speed.

Understanding velocity helps predict the motion dynamics of the spring, showing when it speeds up and slows down through its oscillation.

Trigonometric functions, like sine and cosine, play a vital role in describing harmonic motion. These functions are periodic, making them ideal for modeling repetitive motions like that of a spring. The cosine function in the given displacement equation, \(x = 10 \cos t\), describes how the displacement varies over time.

Cosine functions periodically oscillate between -1 and 1, capturing the full range from the spring's extreme positions. The amplitude of 10 indicates the maximum displacement the spring achieves.

• The displacement starts at maximum when the cosine function is \(1\).
• As time progresses, the displacement decreases as \(\cos t\) moves towards \(0\).
• Eventually, it switches into negative values as \(\cos t\) becomes negative, indicating the spring has passed the equilibrium position into the opposite direction.

Similarly, the sine function, involved in the velocity calculation \(v(t) = -10 \sin t\), shows us when the spring reaches its peak speed and when it's momentarily at rest. Sine's nature of starting at zero complements cosine, beginning where cosine has its maximum or minimum.

For those studying oscillations, understanding how trigonometric functions govern movement is key. They explain everything from the spring's position to the intricacies of its speed, forming a foundation for analyzing many types of periodic motions.