A thin rectangular magnet suspended freely has a period of oscillation equal to t. Now it is broken into two equal halves(each having half of the original length) and one piece is made to oscillate freely in the same field. If its period of oscillation is t', the ratio t'/t is :-
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A thin rectangular magnet suspended freely has a period of oscillation equal to T. Now it is broken into two equal halves (each having half of the original length) and one piece is made to oscillate freely in the same field. if its period of oscillation is T', the ratio T
′
/T is
Hard
Solution
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Correct option is
B
1/2
Hint: Use formula for time period for oscillating magnet
Step 1: Time period of original magnet
Time period for oscillating magnet is:
T=2π
MB
I
where, Magnetic Moment M=xl ( x is the pole strength and l is length)
Moment of Inertia, I=
12
M
′
L
2
Step 2: Time period of one part of magnet
When the magnet is broken in two equal parts, the new magnetic dipole moment is
M
new
=
2
x×l
=
2
M
=> I
new
=
12
(M
′
/2)×(L/2)
2
=
12×8
M
′
L
2
=
8
I
∴ New Time Period is
T
new
= 2π
M
new
B
I
new
Step 3: Ratio of Time period in both cases
Taking the ratio of T
new
and T we get :
T
T
new
=
M
new
I
new
×
I
M
=
I
I
new
×
M
new
M
=
8
1
×
1
2
=
2
1