A small boat is crossing a lake on a windy day. During some interval of time, the boat undergoes the given displacement Δ → r . Δ → r = ( 3.29 m ) ^ ı + ( 2.67 m ) ^ ȷ During the same interval of time, the wind exerts the given constant force → F on the boat. → F = ( 281 N ) ^ ı − ( 135 N ) ^ ȷ What is the total work done on the boat by the wind over this period of time?

564.04Nm

Explanation:

Work is said to be dome when an applied force on an object cause the object to move through a distance. Mathematically;

Work done = Force × Distance in the direction of the force

Given the displacement

r = (3.29 m)^ ı + (2.67 m ) ^ ȷ

Force F = ( 281 N ) ^ ı − ( 135 N ) ^ ȷ

Vectorially, i.i = j.j = 1 and i.j = 0(dot product of different components gives 0)

Work done F = {(281 N) ^ ı − (135N)^ȷ} × {(3.29 m)^ı + (2.67 m )^ȷ}

Multiplying both vectors will give;

= 281i(3.29i) + 281i(2.67j) - 135j(3.29i) - 135j(2.67j)

Since i.j= 0 and i.I = j.j = 1, the answer becomes;

= 924.49(i.i)-360.45(j.j)

= 924.49-360.45

= 564.04Nm

The total work done is 564.04Nm

The dot product of two vector quantities, is given by the sum of the product of the corresponding or parallel components

The total work done on the boat by the wind is 564.04 Joules

The reason the value given for the total work is correct is as follows

The given parameters are:

The displacement the boat undergoes is

The constant force exerted by the wind,

The required parameter:

The total work done on the boat by the wind over this period of time

Solution:

The work done by a force (vector) over a distance (vector) is given by the dot product of the two vectors as follows;

, therefore;

The total work done on the boat by the wind, , is given as follows;

= (281 N × 3.29 m + (-135 N) × 2.67 m) = 564.04 J

The total work done on the boat by the wind, = 564.04 Joules

Learn more about the work done by a force here:

link

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