Question

A fisherman in a boat is using a ’10 lb test’ fishing line. This means that the line can exert a force of 45 N without breaking$\left(\mathbf{1}\mathbf{lb}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{45}\mathbf{N}\right)$. (a) How heavy a fish can the fisherman land if he pulls the fish upward vertically at a constant speed? (b) If he accelerates the fish upward at$\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{m}{\mathbf{/}\mathbf{s}}^{\mathbf{2}}$, what heaviest fish can he land? (c) Is it possible to land a 15 lb trout on a 10 lb test line? Why or why not?

Short answer

(a) The weight of the heaviest fish that can be pulled out of the water is $45\mathrm{N}$or $10\mathrm{lb}$.

(b) The weight of the fish is $37\mathrm{N}$.

(c) No, it is not possibleto land a 15 lb trout on a 10 lb test line.

Step-by-step solution

Step 1. Newton’s second law

Newton's second law states that the force acting on an object causes a change in the object's momentum.

For a constant velocity of the object, the value of the force on that object will be zero.

Step 2. Given information

Given data:

The force exerted by the fishing line is $mg=45\mathrm{N}$.

The acceleration is$a=2.0\mathrm{m}{/\mathrm{s}}^2$.

Step 3. Calculate the weight of the heaviest fish that can be pulled out of the water

(a)

Draw a free-body diagram.

Here, $F_T$is the tension in the rope.

In this case, the speed is constant. So, the acceleration will be $a=0$.

Applying the equilibrium condition along the vertical direction,

$\begin{array}{c}\sum F_y=F_T-mg\\ ma=F_T-mg\\ F_T=m\left(a+g\right)\end{array}$

Substituting the value of acceleration in the above expression,

$\begin{array}{c}{F}_T=m\left(0+g\right)\\ F_T=mg\\ F_T=\left(45\mathrm{N}\right)\left(\frac{1\mathrm{lb}}{4.45\mathrm{N}}\right)\\ F_T=10\mathrm{lb}\end{array}$

Thus, the weight of the heaviest fish that can be pulled out of the water is $45\mathrm{N}$or $10\mathrm{lb}$.

Step 4. Calculate the weight of the heaviest fish

(b)

The mass of the fish can be calculated by using Newton’s second law.

$\begin{array}{c}{F}_{T,\max }-mg=ma\\ m=\frac{F_T}{g+a}\\ m=\frac{\left(45\mathrm{N}\right)}{\left(9.8\mathrm{m}{/\mathrm{s}}^2\right)+\left(2.0\mathrm{m}{/\mathrm{s}}^2\right)}\\ m=3.8\mathrm{kg}\end{array}$

The weight of the fish can be calculated as

$\begin{array}{l}{W}_F=mg\\ W_F=\left(3.8\mathrm{kg}\right)\left(9.8\mathrm{m}{/\mathrm{s}}^2\right)\\ W_F=37\mathrm{N}\end{array}$

Thus, the weight of the fish is $37\mathrm{N}$.

Step 5. Calculate the weight of the heaviest fish

(c)

The mass of the trout is 15 lb. It is not possible to hold a 15 lb trout (fish) using a 10 lb fishing line. If the fish is reeled in a while still in the water, a net is used to remove the fish from the water.