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Question

The propelling force of a rocket increases uniformly from zero to 5 N in the first 12 m and remains constant for the next 40 m. Find the total work done.


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Solution

Step 1: Given data

From a distance of x=0m to x=12m, force changes from zero to, F=5N

Force remains constant for the next, d=40m

Step 2: Calculating proportionality constant

Using the Hooke's law, it can be said that,

Fx

where, F is force and x is distance.

Therefore,

F=kx

where, k is a proportionality constant.

Substituting F=5N and x=12m in the above equation, we get,

5N=k×12mk=512

Step 3: Calculating work done in the first 12 m

Let the work done in the first 12m be W1. Therefore, work done from a distance of x=0m to x=12m can be calculated as follows:

W1=012Fdx=012kxdx=k012xdx=512×x22012=512×1222-022=512×72=30J

Step 4: Calculating work done for the next 40 m

Work done is given as,

W=Fd

where, F is force and d is displacement.

Let work done in the next 40m be W2.

Now for d=40m, force is F=5N(Since force remains constant)

Therefore, work done will be,

W2=5N×40mW2=200J

Step 5: Calculating total work done

The total work done will be the work done in the first 12m and the work done in the next 40m. Therefore, the total work done will be,

W=W1+W2W=30J+200JW=230J

Therefore, total work done is 230J.


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