Respuesta :
Answer:
a) factor [tex]b=\sqrt{2}[/tex]
b) factor [tex]b=\frac{1}{2}[/tex]
c) factor [tex]b=1[/tex]
d) factor [tex]b=1[/tex]
Explanation:
Time period of oscillating spring-mass system is given as:
[tex]T=\frac{1}{f}[/tex]
[tex]T={2\pi} \sqrt{\frac{m}{k} }[/tex]
where:
[tex]f=[/tex] frequency of oscillation
[tex]m=[/tex] mass of the object attached to the spring
[tex]k=[/tex] stiffness constant of the spring
a) On doubling the mass:
- New mass, [tex]m'=2m[/tex]
Then the new time period:
[tex]T'=2\pi\sqrt{\frac{m'}{k} }[/tex]
[tex]T'=2\pi\sqrt{\frac{2m}{k} }[/tex]
[tex]T'=\sqrt{2}\times 2\pi\sqrt{\frac{m}{k} } }[/tex]
[tex]T'=\sqrt{2} \times T[/tex]
where the factor [tex]b=\sqrt{2}[/tex] as asked in the question.
b) On quadrupling the stiffness constant while other factors are constant:
New stiffness constant, [tex]k'=4k[/tex]
Then the new time period:
[tex]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[/tex]
where the factor [tex]b=\frac{1}{2}[/tex] as asked in the question.
c) On quadrupling the stiffness constant as well as mass:
New stiffness constant, [tex]k'=4k[/tex]
New mas, [tex]m'=4m[/tex]
Then the new time period:
[tex]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[/tex]
where factor [tex]b=1[/tex] as asked in the question.
d) On quadrupling the amplitude there will be no effect on the time period because T is independent of amplitude as we can observe in the equation.
so, factor [tex]b=1[/tex]
(a) The value of factor b is [tex]\sqrt{2}[/tex] for the double value of mass.
(b) The value of factor b is 1/2 for the quadruple value of spring constant.
(c) The value of factor b is 1 for the quadruple value of mass and spring stiffness.
(d) On quadrupling the amplitude there will be no effect on the time period.
The expression for the time period of oscillating spring-mass system is given as,
[tex]T=2 \pi \sqrt{\dfrac{m}{k}}[/tex]
Here, m is the mass of spring and k is the spring constant or spring stiffness.
(a) On doubling the mass, the new mass is, m' = 2m.
Then, the new time period is,
[tex]T'=2 \pi \sqrt{\dfrac{m'}{k}}\\\\T'=2 \pi \sqrt{\dfrac{2m}{k}}\\\\T'=\sqrt{2} \times 2 \pi \sqrt{\dfrac{m}{k}}\\\\T'=\sqrt{2} \times T[/tex]
Thus, we can conclude the value of factor b is [tex]\sqrt{2}[/tex] for the double value of mass.
(b)
With quadruple spring stiffness, the new spring stiffness is, k' = 4k. Then the new time period is,
[tex]T'=2 \pi \sqrt{\dfrac{m}{k'}}\\\\T'=2 \pi \sqrt{\dfrac{2m}{4k}}\\\\T'=\dfrac{1}{2} \times 2 \pi \sqrt{\dfrac{m}{k}}\\\\T'=\dfrac{1}{2} \times T[/tex]
Thus, we can conclude the value of factor b is 1/2 for the quadruple value of spring constant.
(c)
With quadruple mass and spring stiffness, the new mass and spring stiffness are, m' = 4m and k' = 4k. Then the new time period is,
[tex]T'=2 \pi \sqrt{\dfrac{m'}{k'}}\\\\T'=2 \pi \sqrt{\dfrac{4m}{4k}}\\\\T'=1 \times 2 \pi \sqrt{\dfrac{m}{k}}\\\\T'=1 \times T[/tex]
Thus, we can conclude the value of factor b is 1 for the quadruple value of mass and spring stiffness.
(d)
On quadrupling the amplitude there will be no effect on the time period because T is independent of amplitude as we can observe in the equation.
so, factor b =1.
Learn more about the time period of spring mass system here:
https://brainly.com/question/14833462
