Vibrations of a spring Suppose an object of mass m is...

Vibrations of a spring Suppose an object of mass $m$ is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position $y=0$ when the mass hangs at rest. Suppose you push the mass to a position $y_{0}$ units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system , the position y of the mass after t seconds is $$y=y_{0} \cos (t \sqrt{\frac{k}{m}})(4)$$ where $k>0$ is a constant measuring the stiffness of the spring (the larger the value of $k$, the stiffer the spring) and $y$ is positive in the upward direction. Use equation (4) to answer the following questions. a. Find $d y / d t$, the velocity of the mass. Assume $k$ and $m$ are constant. b. How would the velocity be affected if the experiment were repeated with four times the mass on the end of the spring? c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness ( $k$ is increased by a factor of 4 )? d. Assume y has units of meters, $t$ has units of seconds, $m$ has units of kg, and $k$ has units of $\mathrm{kg} / \mathrm{s}^{2} .$ Show that the units of the velocity in part (a) are consistent.

Vibrations of a spring Suppose an object of mass $m$ is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position $y=0$ when the mass hangs at rest. Suppose you push the mass to a position $y_{0}$ units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system , the position y of the mass after t seconds is
$$y=y_{0} \cos (t \sqrt{\frac{k}{m}})(4)$$
where $k>0$ is a constant measuring the stiffness of the spring (the larger the value of $k$, the stiffer the spring) and $y$ is positive in the upward direction.
Use equation (4) to answer the following questions.
a. Find $d y / d t$, the velocity of the mass. Assume $k$ and $m$ are constant.
b. How would the velocity be affected if the experiment were repeated with four times the mass on the end of the spring?
c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness ( $k$ is increased by a factor of 4 )?
d. Assume y has units of meters, $t$ has units of seconds, $m$ has units of kg, and $k$ has units of $\mathrm{kg} / \mathrm{s}^{2} .$ Show that the units of the velocity in part (a) are consistent.

Key Concepts

Dimensional Analysis

Dimensional analysis is a technique used to verify the consistency of units in physical equations. It involves checking that both sides of an equation or a derived physical quantity have the same fundamental units, ensuring that the equation is physically meaningful. In the context of a spring-mass system, this helps demonstrate that the computed velocity from the derivative has the correct units of distance over time.

Simple Harmonic Motion

This concept deals with periodic motion, where an object oscillates about an equilibrium position under a restoring force proportional to displacement. In an oscillating spring?mass system, the motion follows a sinusoidal pattern, reflecting the characteristics of simple harmonic motion, and is described by equations that feature sine or cosine functions.

Differentiation and the Chain Rule

Differentiation is a fundamental operation in calculus used to determine the rate at which a quantity changes. When dealing with composite functions, such as the cosine of a function of time, the chain rule is applied to differentiate the outer function with respect to its inner function, and then multiplied by the derivative of the inner function.

Angular Frequency

Angular frequency is a measure of how many radians per second an oscillating object completes and is crucial in describing the dynamics of oscillatory systems. In a spring-mass system, angular frequency is typically given by the square root of the ratio of the spring constant to the mass, reflecting how both mass and spring stiffness influence the speed of oscillations.

Effects of Mass and Stiffness on Oscillatory Motion

Changes in mass and spring stiffness affect the oscillatory motion by altering the angular frequency. Increasing the mass tends to slow down the oscillation, while increasing the spring stiffness results in a faster oscillation. These effects are described mathematically by the angular frequency formula, making it clear how each parameter influences the rate of oscillation.

Transcript

00:05 For part a of the given problem, we need to find the derivative first.

00:10 So the given function is y equals y -not cost of t under root k over m.

00:26 In order to find the derivative, we need to apply the product rule.

00:32 So, d -y by d -t becomes d -t -of -y -0 into cos of t under root k over m plus cost of t under root k over m into y -not.

01:02 Since y not is assumed to be a constant, this expression becomes zero.

01:11 Thereby, we are left with d by dt of cost of t under root k over m into y not.

01:29 Now applying chain rule.

01:33 We can assume the inner function u.

01:36 To be equal to t into under root k over m and the after function y equals cause of you.

01:47 Finding the derivative of each of these equations, we get d u by dt equals under root k over m, since k and m both are constants.

02:06 D .y by d u becomes minus 6.

02:10 Of u.

02:12 Hence the equation for d y by d t becomes y not under root k over m into minus sign of u next we simplify and put in the value of u we get minus y not under root k over m sign of t under root k over m for part b we are to find how increasing mass four times affect the derivative thus putting m equals 4m in the equation of derivative you've just evaluated we get d .y by dt equals negative y not under root k over 4m, sine of t under root k over m, 4m.

03:28 On simplification, we get 1 by 2, y not under root k over m, into sign of 1 by 2t under root k over m.

03:45 This implies that the velocity will be half of what originally was, whereas the period is going to be twice of the original...

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