The Linear Spring-Mass System

Classical Mechanics Level 5

A Mass $\displaystyle m$ is attached to a spring of mass $\displaystyle M$ and spring constant $\displaystyle K$ .

If the velocity of the mass $\displaystyle m$ varies linearly over the length of the spring when it is given a kick, Let $\displaystyle T$ denote the Time Period of the system.

Find $\displaystyle 1000T$ to the nearest integer.

Details and Assumptions:
$\bullet m = 3Kg$
$\bullet M = 0.5Kg$
$\bullet K = 100N/m$

The answer is 1118.

2 solutions

Mandar Sohoni
Feb 26, 2014

The effective mass of the spring used in calculating the time period is M/3. So the time period is 2(pi)((M/3 + m)/K)^(1/2) instead of 2(pi)(m/k)^(1/2) Plugging in the values, you get T=1.11810... =>1000T = 1118.10... To find out how the effective mass is M/3, you can find the kinetic energy if the spring. Consider an element of the spring of length dx at a distance x from the wall, and let the other end of the spring move with a velocity v. Now dx moves with a velocity of vx/L (the total length of the spring being L). The mass dM of the element is Mdx/L. Therefore the kinetic energy of dx, dKE = (1/2)(Mdx/L)(vx/L)^2. Now, integrating from 0 to L, we end up with KE = (1/6)(M)(v)^2 = (1/2)(M/3)(v)^2

how does dx move with a velocity vx/L????....

Sayam Chakravarty - 7 years, 3 months ago

Since the velocity of the spring varies linearly, the velocity of any elemental particle at a distance x from the fixed end can be given as $u= ax + b$ , where $a$ and $b$ are constants which can be easily determined.

At $x=0$ , i.e, at the fixed end $u=0$ . Therefore, $0 = 0 + b \Rightarrow b=0$ .

Similarly, at $x=L$ , i.e, at the point connected to the block $u=v$ . Therefore, $v= aL \Rightarrow a=\frac{v}{L}$ .

Substituting the above values in the expression, we get, $u=\frac{vx}{L}$ .

Sudeep Salgia - 7 years, 2 months ago

thanks

Sayam Chakravarty - 7 years, 2 months ago

THANK! ! I was really in need of solution to this type of question .....

Aniket Sanghi - 5 years, 2 months ago

Jaivir Singh
Apr 27, 2014

T = 2PIE [ ROOT OF (m+M/3) / K ]

The Linear Spring-Mass System

The answer is 1118.

2 solutions

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