An object of unknown mass oscillates on the end of a spring wit period 8 s. A 10kg object is attached to the first object, changing te period to 12 s. What is the mass of the first object

Explanation:

Let the first object have a mass of M

And a period of T1=8sec

The second object has a mass 10kg and a period of T2=12 sec

It is know that,

The period of a spring-mass system is proportional to the square root of the mass and inversely proportional to the square root of the spring constant.

T=2π√(m/k)

Then the constant in this equation is the spring constant (k) and 2π, which does not change for the same material.

Then, make k subject of formulas

T²=4π²(m/k)

T²k=4π²m

Then, k/4π²=m/T²

So the k is directly proportional to m and inversely proportional to T²

M1/T1²=M2/T2²

Since, M1 is unknown, M2=10kg, T1=8sec and T2=12

Then,

M1/T1²=M2/T2²

M1/8²=10/12²

M1/64=0.06944

M1=0.06944×64

M1=4.444kg

The mass of the first object is 4.44kg

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