BITSAT 2012 Paper

Step 1: Calculation of spring constant of wire
Young's modulus of wire, Y=‌

∆F∕A

∆L∕L

=‌

∆F

∆L

×‌

L

A

Due to elasticity of the wire, it behaves as equivalent spring (K‌spring ‌=F∕∆x). Hence, Spring constant for Wire, KW=‌

∆F

∆L

=Y‌

A

L

Step 2: Calculation of Equivalent spring constant
Let a force F be applied to the end of the spring.
The wire will extend.
Total extension ∆S=x‌wire ‌+x‌spring ‌=‌

Fw

Kw

+‌

FS

K

But FS=Fw‌‌ (By Newton's, third Law, Both will apply equal and opposite forces at their joint A)
∴∆S=Fw(‌

1

Kw

+‌

1

K

)
So, Equivalent spring constant (Keq) is given by:
Keq=‌

Feq

Xeq

=‌

Fw

∆S

=‌

1

1 Kw +‌ 1 K

Keq=‌

(K)(Kw)

Kw+K

=‌

KY‌ A L

Y‌ A L +K

=‌

KYA

YA+KL

Step 3: Calculation of Time period
Angular frequency ω for spring block system is given by:
ω=√‌

Keq

m

=√‌

KYA

m(YA+KL)

∴‌ Time period, ‌T=‌

2π

ω

=2π√‌

(YA+KL)m

KYA