The function that describes this oscillation in time is of the form Y (t) = eat (2 cos(bt) + sin(bt)), where x a + bi is the solution with b > 0 to the following equation mx + rx+ k = 0. In this equation, m represents the mass in kg, k is the spring constant and measures the spring's elasticity, r measures the friction damping effects. (a) Suppose the mass weighs 10 kg, the spring constant k is 2 and r is 4. Find a and b by solving the quadratic equation. (b) Substitute a and b in the expression for Y(t). (c) Find the initial extension of the spring A = Y(0).
The function that describes this oscillation in time is of the form Y (t) = eat (2 cos(bt) + sin(bt)), where x a + bi is the solution with b > 0 to the following equation mx + rx+ k = 0. In this equation, m represents the mass in kg, k is the spring constant and measures the spring's elasticity, r measures the friction damping effects. (a) Suppose the mass weighs 10 kg, the spring constant k is 2 and r is 4. Find a and b by solving the quadratic equation. (b) Substitute a and b in the expression for Y(t). (c) Find the initial extension of the spring A = Y(0).
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Transcribed Image Text:/Assignment3.pdf
Y=-A
Figure 1: A mass-spring system.
Reference: https://upload.wikimedia.org/wikipedia/commons/8/8d/Oscillation-terms.svg.
The function that describes this oscillation in time is of the form
Y (t) = eat (2 cos(bt) + sin(bt)),
where x = a + bi is the solution with b > 0 to the following equation
mx? + rx + k = 0.
In this equation, m represents the mass in kg, k is the spring constant and measures the
spring's elasticity, r measures the friction damping effects.
(a) Suppose the mass weighs 10 kg, the spring constant k is 2 and r is 4.
Find a and b by solving the quadratic equation.
(b) Substitute a and b in the expression for Y(t).
(c) Find the initial extension of the spring A = Y(0).
(d) Plot Y(t) for positive times. You are allowed to use any graphical software including
Wolfram Alpha for this purpose.
(e) What do you observe? What will happen for a very large t?
4. Population dynamics.
In the ski resort, the amount of people visiting changes every day of the year. In the cold
o neonle coming, while in the warm period very few are visiting.
Transcribed Image Text:3. Dynamics of a spring.
Consider the behaviour of a mass-spring system (see Figure 1 below). If it is at equilib-
rium, it remains in a rest position. Once it receives a force, it will start oscillating. In this
problem, we will study the dynamics of these oscillations. We will take as reference the center
of mass and we consider Y = 0 when the system is at rest. We then compress the spring to
Y = A and study the behaviour of Y in time.
I
Y=A
Y=0
-A
Y=-A
Figure 1: A mass-spring system.
Reference: https://upload.wikimedia.org/wikipedia/commons/8/8d/Oscillation-terms.svg.
The function that describes this oscillation in time is of the form
Y (t) = et (2 cos(bt) + sin(bt)),
where a = a + bi is the solution with b> 0 to the following equation
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