Question Video: Calculating the Amount of Work Done by a Variable Force with an Unknown Constant | Nagwa Question Video: Calculating the Amount of Work Done by a Variable Force with an Unknown Constant | Nagwa

Question Video: Calculating the Amount of Work Done by a Variable Force with an Unknown Constant Mathematics • Third Year of Secondary School

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A block moves in a straight line under the action of a force 𝐹 = (12𝑠² + 6𝑠 + 𝑐) N, where 𝑠 meters is the displacement of the body from its initial position. The work done by the force in moving the block from 𝑠 = 0 m to 𝑠 = 3 m is 34 J. Determine the work done by 𝐹 in moving the block from 𝑠 = 3 m to 𝑠 = 6 m.

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Video Transcript

A block moves in a straight line under the action of a force 𝐅 equal to 12𝑠 squared plus six 𝑠 plus 𝑐 newtons, where 𝑠 meters is the displacement of the body from its initial position. The work done by the force in moving the block from 𝑠 equals zero meters to 𝑠 equals three meters is 34 joules. Determine the work done by 𝐅 in moving the block from 𝑠 equals three meters to 𝑠 equals six meters.

As we have a force acting on an object moving in a straight line, we can use the formula 𝑊 is equal to the integral of 𝐅 with respect to 𝑠 to calculate the work done by the force. In this question, we are told that the force 𝐅 is equal to 12𝑠 squared plus six 𝑠 plus 𝑐 newtons. The work done is therefore equal to the integral of this with respect to 𝑠. Integrating each term with respect to 𝑠, we have four 𝑠 cubed plus three 𝑠 squared plus 𝑐𝑠. We are told that the work done by the force in moving the block from 𝑠 equals zero meters to 𝑠 equals three meters is 34 joules. We can therefore substitute in these values as shown as this will enable us to calculate the constant 𝑐.

When 𝑠 equals three, the right-hand side of our equation becomes 135 plus three 𝑐. And when 𝑠 equals zero, this expression equals zero. This means that 34 is equal to 135 plus three 𝑐. Subtracting 135 from both sides of this equation, we have three 𝑐 is equal to negative 101. We can then divide through by three such that 𝑐 is equal to negative 101 over three. Substituting this back in to the expression for the work done, we have 𝑊 is equal to four 𝑠 cubed plus three 𝑠 squared minus 101 over three 𝑠.

As we need to calculate the work done by 𝐅 from 𝑠 equals three to 𝑠 equals six, we can substitute these values into our expression. When 𝑠 equals six 𝑊 is equal to 770, and when 𝑠 equals three, 𝑊 is equal to 34. The work done from 𝑠 equals three meters to 𝑠 equals six meters is therefore equal to 770 minus 34, which is equal to 736. The final answer is equal to 736 joules.

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