Two tugboats pull a disabled supertanker. Each tug exerts a

constant force of 1.90×106 N, one at an angle 19.0 west of north,

and the other at an angle 19.0 east of north, as they pull the

tanker a distance 0.870 km toward the north. What is the total work done by the two tugboats on the

supertanker? Express your answer in joules, to three significant figures.

General guidance

Concepts and reason

The concepts required to answer the problem are resolution of forces and work done due to a force.

The direction of forces applied is given. The forces can be resolved into its components and the net force can be calculated. Also, as the displacement of the supertanker is given, the work done by the tugboats can be calculated.

Fundamentals

The forces are said to be balanced, when the net force along a specific direction is zero.

Resolution of forces is the splitting of a forces into its different components which are perpendicular to each other.

The Force F can be split into two components as shown

Here is the net force, is the horizontal component of force, is the vertical component of force

Here is the force and is the angle made by the force with axis.

Work is said to be done when a force acts on body and displaces it by some distance.

It is given by

Here, is the force, is the displacement and is the angle between force vector and displacement vector.

Its unit is Joules and is denoted by J.

Step-by-step

Step 1 of 2

The given forces can be resolved as follow

Since, the force’s component towards East is equal to the force’s components towards west, the force along East – West line will be balanced.

The other component of forces lies in the same direction, towards the North. Therefore, net force will be towards the north.

Net force is towards north is given by

Substitute for

The two forces due to tugboats is split into its components, along North-South and East- West line. It is observed that, the East-West components of both the forces are equal and opposite. Therefore, forces along East-West line are balanced. The components along North- South line are equal and in the same direction (towards North), therefore, the net force will be towards North and will be equal to sum of both the components. The expression for net force was derived and further the values were substituted.

It is important to take care while splitting the force into its components. The component of force along which angle is given is always .

Use the force calculated in the previous step to determine the work done due to this force.

Step 2 of 2

Work done is given by

Substitute for , for and for

The work done by the tugboats is

The magnitude of net force was calculated in the previous part and its direction was towards the North. The displacement of the supertanker is given and its direction is also towards the North. Therefore, angle between the net force and displacement is 00. The value of net force and displacement was substituted in the work done formula and the total work done by the tugboats was calculated.

It is important to note that the net force due to the tugboats acts in the same direction as the displacement. Therefore, the angle between them is and not 1800. Also, while substituting the magnitude of displacement, the units should be converted into meter.

Answer

The work done by the tugboats is

Answer only

The work done by the tugboats is

F Fsin e Fcose D

F = F +F,

F

х

у

F F cos0Fsin0

F

Fd cos 0 W.D

F

North 2F cos19 F F 19°19° West F sin 190 East Fsin 19 LL

Е – 2F cos19° пet

1.90x 10 N

F

Fn – 2(190×10’N) соs19° .90×106 Е -3.59х10°N пеt

F cos

W.D Fd cos 0

3.59×10 N

F

0.870×10’m

00

WD-(3.59×10 N) (0.870x10m)cos 0

wD-(3.59×10 N) (0.870x10m) W.D 3.12×10’j

00

W.D 3.12X10°J

W.D 3.12X10°J

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