Question

a spring of spring constant k is cut into three parts of length in ratio1:2:1.the algebric addition of spring constants of these springs will be

Answer 1

Answer:

10k

Explanation:

The formula for spring constant k=E xA / L where E is the Young's modulus of the material A the sectional area, and L is the length of the ideal spring.

It means that spring constant is inversely related to length.

Now if cut the spring into three parts, individual parts would have different spring constants which would be;

Length of A = 1/4L, means its K would be four times now

Length of B = 2/4L, means its K would be two times now

Length of AC= 1/4L, means its K would be four times now

Now if we sum all these, their sum would be 10 times the original spring constant

Answer 2

The algebraic addition of spring constants will be 10K

Step by step explanation:

$\textbf{\Large The formula of Young's modulus :}\gamma = \frac{\textbf{\Large K L}}{\textbf{\Large A}} \\\\\textbf{\large Where K = Spring constant, A = Area of cross section}\\\\\textbf{\large L = Length of spring}$

So spring constant: K = $\frac{\gamma \times A }{L}$                                     (Equation 1)

As Young's modulus γ  and  Area A of spring remains constant on cutting the spring.

So only length of the spring changed.

The spring is cut into three parts of length in ratio 1 : 2 : 1 (given)

By this ratio, If the length of original spring was say, L

Then on cutting the spring,

The length of first spring = $\frac{L}{4}$

The length of second spring =$\frac{L}{2}$

The length of third spring =  $\frac{L}{4}$

γ   and A are constant

(Putting the values in equation 1)

The spring constant of first spring =  $K_1 = \frac{\gamma \times A }{\frac{L}{4} } = 4K$

The spring constant of second spring = $K_2 = \frac{\gamma \times A }{\frac{L}{2} } = 2K$

The spring constant of third spring = $K_3 = \frac{\gamma \times A }{\frac{L}{4} } = 4K$       (because K = $\frac{\gamma \times A }{L}$ )

$\textbf{\Large So the algebraic addition of spring constants will be = }\\\\\textbf{\Large 4K + 2K + 4K = 10K}$