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A spring has an unstretched length of 10 cm. It exerts a restoring force F when stretched to a length of 11 cm. a. For what length of the spring is its restoring force 3F? b. At what compressed length is the restoring force 2F?

Respuesta :


x₀ = unstretched length of spring = 10 cm

x' = stretched length = 11 cm

k = spring constant of the spring

F = restoring force when stretched length is x'

restoring force when stretched length is x'  is given as

F = k (x' - x₀)

F = k (11 - 10)

F = k


(a)

x'' = stretched length

F'' = restoring force when stretched length is x'' = 3F

restoring force when stretched length is x'' is given as

F'' = k (x'' - x₀)

3F = F (x'' - 10)

x''= 13 cm


(b)

x''' = compressed length

F''' = restoring force when compressed length is x'' = 2F

restoring force when  when compressed length is x'' is given as

F''' = k (x₀ - x''' )

2F = F (10 - x''' )

x''' = 8 cm


fichoh

Using the spring constant relation, the required length and compressed length of the spring at the given force values are 13 cm and 8 cm respectively.

Recall :

  • F = ke
  • k = spring constant ; e = extension
  • e = (Final length - Initial length)

Initial length, l = 10

Final length = L1 = 11

F = k(11 - 10)

F = k(1) ;

F = k

Length for a restoring force of 3F ;

3F = k(Final length - Initial length)

k = F

3F = F(L1 - 10)

3F = - 10F + F(L1)

F(L1) = 3F + 10F

F(L1) = 13F

L1 = 13

Hence, required length ls 13 cm

B.)

For compressed length :

Let compressed length = c

2F = F(initial length - compressed length)

2F = F(10 - c)

2F = 10F - F(c)

F(c) = 10F - 2F

F(c) = 8F

c = 8F / F

c = 8 cm

Therefore, the required compressed length ls 8 cm

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