I am having trouble with a couple multiple choice problems for Physics. Please h

Question

I am having trouble with a couple multiple choice problems for Physics. Please help!

#1). An object of mass M suspended by a spring vibrates with period T. If this object is replaced by one of mass 9.00 M, the new object vibrates with a period:   

#2) An object with mass M suspended by a spring vibrates with frequency f. When a second object is attached to the first, the system now vibrates with frequency f/2.00. The mass of the second object is:

The mass of the second object is

#3) A pendulum on earth swings with angular frequency ?. On an unknown planet, it swings with angular frequency ?/ 2.00. The acceleration due to gravity on this planet is:

A pendulum on earth swings with angular frequency  On an unknown planet, it swings with angular frequency  2.00. The acceleration due to gravity on this planet is

#4) A mass oscillates with simple harmonic motion of amplitude A. The kinetic energy of the mass will equal third of the potential energy of the spring when the position is:

Thank you!

Explanation / Answer

1)
we know time peroid equation for spring mass system is
T =2pi*sqrt(m/k)
if mass is replaced by 9M then
time period wii change to T1 = 3T
option B)
-------------------------------
2)
f = (1/2pi)*sqrt(k/M)
if frequency is f/2
then new mass will be
4M
mass added is M+3M
mass of the second object is
3M
option B)
--------------------------------
3)
angular frequency = sqrt(g/l)
as frequency decreases by half so acceleration due to gravity will be
4g
option
A) 4g
--------------------------------
4)
kinetic energy K= 0.5mw^2(A^2-x^2)
potential energy U= 0.5mw^2x^2
as per given condition
K = U/3
0.5mw^2(A^2-x^2) = (0.5mw^2x^2)/3
3A^2-3x^2 = x^2
3A^2 = 4x^2
Amplitude A = 2x/sqrt(3).

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