Question

Two ropes are connected to a 200 kg dinghy. Two brothers each take one rope and pull. When the brothers pull in the same direction, the dinghy accelerates at a rate of 1.33 m/s2 to the east. If they pull in opposite directions, the dinghy has an acceleration of 0.514 m/s2 to the west. Assume the ropes are horizontal, and ignore any other horizontal forces acting on the dinghy..

a) What is the magnitude of the force each brother exerts on the dinghy? (Assume each magnitude is constant in either direction.)

larger force _________________ N

smaller force _________________ N

Answer 1

Answer:

184.4 N, 81.6 N

Explanation:

To solve this problem, we can use Newton's second law of motion, which states that the net force on an object is equal to the product between the mass of the object and its acceleration:

$\sum F=ma$

where

$\sum F$ is the net force

m is the mass

a is the acceleration

In this problem:

m = 200 kg is the mass of the dinghy

At the beginning, the two brothers pull in the same direction, so we can call the net force

$\sum F=F_1+F_2$

where $F_1,F_2$ are the forces exerted by the two brothers. So, Newton's second law in this case becomes:

$F_1+F_2=ma_1$ (1)

where

$a_1=1.33 m/s^2$ is the acceleration in the 1st case (to the east)

When the two brothers pull in opposite directions,

$\sum F=F_1-F_2$

So the equation is

$F_1-F_2=ma_2$ (2)

where $a_2=-0.514 m/s^2$ (negative because it is to the west)

Combining the two equations, we can find the magnitude of the two  forces:

$F_1+F_2=ma_1\\F_1-F_2=ma_2$

From (1),

$F_1=ma_1-F_2$

Substituting into (2),

$ma_1-2F_2=ma_2\\F_2=\frac{a_1-a_2}{2}=\frac{200(1.33-(-0.514))}{2}=184.4 N$

And so, the other force is

$F_1=ma_1-F_2=(200)(1.33)-184.4=81.6 N$

Answer 2

Answer:

Explanation:

mass of dinghy, m = 200 kg

Let the force of pull of one brother is F1 and of another brother is F2.

Case 1: When pull is in the same direction

F1 + F2 = m x a

F1 + F2 = 200 x 1.33

F1 + F2 = 266 .... (1)

Case 2:

F1 - F2 = - 200 x 0.514  

F1 - F2 = - 102.8 ...... (2)

Add both the equations

2F1 = 163.2

F1 = 81.6 N

Put in equation (1)

81.6 + F2 = 266

F2 = 184.4 N

So, larger force is 184.4 N

smaller force is 81.6 N