Question

A mass attached to the end of a long spring is bouncing up and down. As it bounces, its distance from the floor varies sinusoidally with time. When the mass is released, it takes \(0.3 \mathrm{s}\) to reach a high point of 60 cm above the floor. It takes 1.8 s for the mass to reach the first low point of \(40 \mathrm{cm}\) above the floor. a) Sketch the graph of this sinusoidal function. b) Determine the equation for the distance from the floor as a function of time. c) What is the distance from the floor when the stopwatch reads \(17.2 \mathrm{s?}\) d) What is the first positive value of time when the mass is \(59 \mathrm{cm}\) above the floor?

Short answer

a) Sinusoidal graph b) y = 10 * sin(2π/3 * (t - 0.3)) + 50 c) D ≈ 46 cm d) t ≈ 0.635 s

Step-by-step solution

Understand the problem

Identify the key details: the mass takes 0.3s to reach the highest point (60 cm) and 1.8s to reach the first low point (40 cm). This gives a clue that the oscillation is sinusoidal.

Determine the period

The total time to go from the highest point to the lowest point and back is the period. From the given data, the period can be calculated as 2 times the time interval between successive high and low points.The time interval = 1.8s - 0.3s = 1.5s.So the period, T = 2 * 1.5s = 3s.

Find the amplitude

The amplitude is half the difference between the maximum and minimum values of the sinusoidal function.Amplitude, A = (60 cm - 40 cm) / 2 = 10 cm.

Find the vertical shift

The vertical shift, D, is the average of the maximum and minimum values.D = (60 cm + 40 cm) / 2 = 50 cm.

Develop the equation

Using the general form for a sinusoidal function, y = A * sin(B(t - C)) + D:A = 10 cm,D = 50 cm.The period T = 3s, so B = 2π/T = 2π/3.Since the mass reaches the highest point at t = 0.3s, we can use this information to find the phase shift, C. C = -0.3s.The equation becomes:y = 10 * sin(2π/3 * (t - 0.3)) + 50.

Evaluate for d = 17.2s

Using the equation y = 10 * sin(2π/3 * (17.2 - 0.3)) + 50, solve for y.Simplifying: y = 10 * sin(2π/3 * 16.9) + 50.Calculate approximate value of sin(2π/3 * 16.9) and substitute back to find y.

First positive value for y = 59 cm

Solve the equation 59 = 10 * sin(2π/3 * (t - 0.3)) + 50 for t.First, isolate the sine function: 9 = 10 * sin(2π/3 * (t - 0.3)).So, sin(2π/3 * (t - 0.3)) = 0.9.Use the inverse sine function to find the angle and solve for t.

Key concepts

Sinusoidal motion

Sinusoidal motion refers to the smooth repetitive oscillations that can be described by sine or cosine functions. These movements are characterized by their wave-like patterns, which are common in various natural and mechanical processes.

The mass bouncing up and down on a spring is a classic example of sinusoidal motion. It moves in a repetitive wave pattern as it oscillates between its highest and lowest points.

In this problem, we observe that the mass reaches a highest point of 60 cm, then moves down to a lowest point of 40 cm, creating a wave-like motion that we can model using a sinusoidal function.

Amplitude

Amplitude is a measure of how far an oscillating object moves from its equilibrium position. It represents the 'height' of the wave in sinusoidal motion.

To find the amplitude in this problem, we take half the difference between the highest and lowest points reached by the mass. Mathematically, it's calculated as:

Amplitude, A = \(\frac{60 \, \text{cm} - 40 \, \text{cm}}{2}\) = 10 cm.

This value means the mass moves 10 cm above and below its average position (vertical shift) of 50 cm.

Period

The period of a sinusoidal function is the time it takes for one complete cycle of motion. It can be understood as the time to go from the highest point to the lowest point and back to the highest point.

In this case, the mass takes 0.3 seconds to reach the highest point and 1.8 seconds to reach the lowest point. The period (T) can be calculated by doubling the interval from high to low:

Period, T = 2 * (1.8s - 0.3s) = 2 * 1.5s = 3s.

Therefore, one full cycle of the mass oscillating takes 3 seconds.

Vertical shift

Vertical shift refers to how far the central, average position of the sinusoidal function is moved up or down from the origin (zero point). It’s essentially the midpoint between the maximum and minimum values of the vertical position.

For this problem, the vertical shift (D) is calculated as the average of the maximum and minimum heights:

Vertical shift, D = \(\frac{60 \, \text{cm} + 40 \, \text{cm}}{2}\) = 50 cm.

So, the equation that models the distance from the floor over time for this sinusoidal function is:

y = 10 * sin\(\frac{2\pi}{3}\) (t - 0.3) + 50.

This function incorporates the amplitude, period, and vertical shift, accurately describing the mass's motion.