#### To determine

**(a)**

**To find:**

The work needed to stretch the spring from *35 cm* to *40 cm*

#### Answer

*1.04 J*

#### Explanation

**1) Concept:**

Use Hook’s law and formula of work done

**2) Law and Formula:**

Hook’s law:

The force required to maintain a spring stretched *x* units beyond its natural length is proportional to *x,* *fx=kx* Where *k* is a positive constant called the spring constant

Work done:

*W=∫abf(x)dx*

**3) Calculation:**

a) According to Hook’s law, the force required to hold the spring stretched *x* meters beyond its natural length is *fx=kx*

From the given information, *2 J* of work is needed to stretch a spring from its natural length of *30 cm* to length of *42 cm*

Therefore, the amount of stretched is *42-30=12 cm=0.12 m*

Thus,

*2=∫00.12kxdx*

Simplify,

*⇒2=kx2200.12*

*⇒2=k(0.12)22-k(0)22*

*⇒2=0.0144k2-0*

*⇒2=0.0072k*

*⇒k=20.0072*

*⇒k≈277.78 N/m*

So, *fx=277.78x*

Thus, the work needed to stretch the spring from *35 cm* to *40 cm* is

The spring is stretched from *35-30=5 cm=0.05 m* to *40-30=10 cm=0.10 m*

*W=∫0.050.10277.78xdx*

Simplify,

*=277.78x220.050.10*

*=277.780.1022-0.0522*

*=277.780.012-0.00252*

*=277.780.00752*

*=138.890.0075*

*=1.041675*

Therefore,

*W≈1.04 J*

**Conclusion:**

The work needed to stretch the spring from *35 cm* to *40 cm* is *1.04 J*

#### To determine

**b)**

**To find:**

How far beyond its natural length will a force of *30 N* keep the spring stretched?

#### Answer

*10.8 cm*

#### Explanation

**1) Concept:**

Use the calculations in part (a)

**2) Given:**

*fx=30 N*

**3) Calculation:**

From part (a),

*fx=277.78x*

Substitute the value of *f(x)*

*⇒30=277.78x*

*⇒x=30277.78*

Simplify,

*⇒x=0.10799 m=10.799 cm≈10.8 cm*

Therefore,

A force of *30 N* will keep the spring *10.8 cm* stretched beyond its natural length

**Conclusion:**

A force of *30 N* will keep the spring *10.8 cm* stretched beyond its natural length