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3 months ago

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a. For a spring-mass oscillator, if you double the mass but keep the stiffness the same, by what numerical factor does the perio

d change? That is, if the original period was T and the new period is b⁢T, what is b? It is useful to write out the expression for the period and ask yourself what would happen if you doubled the mass.b. If, instead, you quadruple the spring stiffness but keep the mass the same, what is the factor b?c. If, instead, you quadruple the mass and also quadruple the spring stiffness, what is the factor b?d. If, instead, you quadruple the amplitude (keeping the original mass and spring stiffness), what is the factor b?

1 answer:

kicyunya [3.2K]3 months ago

7 0

Answer:

a) factor $b=\sqrt{2}$

b) factor $b=\frac{1}{2}$

c) factor $b=1$

d) factor $b=1$

Explanation:

For an oscillating spring-mass system, the time period is expressed as:

$T=\frac{1}{f}$

$T={2\pi} \sqrt{\frac{m}{k} }$

where:

$f=$ represents the frequency of oscillation

$m=$ signifies the mass linked to the spring

$k=$ is the spring's stiffness constant

a) If the mass is doubled:

New mass, $m'=2m$

Thus, the new time period:

$T'=2\pi\sqrt{\frac{m'}{k} }$

$T'=2\pi\sqrt{\frac{2m}{k} }$

$T'=\sqrt{2}\times 2\pi\sqrt{\frac{m}{k} } }$

$T'=\sqrt{2} \times T$

this leads to factor $b=\sqrt{2}$ as per the question.

b) When the stiffness constant is quadrupled, holding other factors constant:

New stiffness constant, $k'=4k$

Thus, the new time period:

$T'=2\pi\sqrt{\frac{m}{k'} }\\\\T'=2\pi\sqrt{\frac{m}{4k} }\\\\T'=\frac{1}{2} \times 2\pi\sqrt{\frac{m}{k} } }\\\\T'=\frac{1}{2} \times T$

this results in factor $b=\frac{1}{2}$ as required.

c) When both mass and stiffness constant are quadrupled:

New stiffness, $k'=4k$

New mass, $m'=4m$

Thus, the new time period:

$T'=2\pi\sqrt{\frac{m'}{k'} }\\\\T'=2\pi\sqrt{\frac{4m}{4k} }\\\\T'=1 \times 2\pi\sqrt{\frac{m}{k} } }\\\\T'=1 \times T$

which leads to factor $b=1$ as stated in the question.

d) If amplitude is quadrupled, the time period remains unaffected because T does not depend on amplitude as demonstrated by the equation.

Thus, factor $b=1$

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