Video Transcript
A particle moves in a straight line
under the action of the force 𝐅, where 𝐅 is equal to sin of 𝜋𝑠 and 𝑠 is
measured in meters. Calculate the work done by the
force 𝐅 when the particle moves from 𝑠 equals zero to 𝑠 equals one-half.
In this question, a variable force
acts on a particle. And both the motion of the particle
and the force acting on it are in one dimension. We can therefore calculate the work
done by the force by using the formula 𝑊 is equal to the integral of 𝐅 with
respect to 𝑠. We are told in the question that
the force 𝐅 is equal to sin of 𝜋𝑠. The work done is therefore equal to
the integral of this with respect to 𝑠. And we need to calculate this
between 𝑠 equals zero and 𝑠 equals one-half. So these are our lower and upper
limits.
We recall that the integral of sin
𝑎𝑥 with respect to 𝑥 is equal to negative one over 𝑎 multiplied by the cos of
𝑎𝑥. This means that our expression
integrates to negative one over 𝜋 multiplied by the cos of 𝜋𝑠. Our next step is to substitute in
our limits. When 𝑠 is equal to one-half, we
have negative one over 𝜋 multiplied by cos of 𝜋 over two. The cos of 𝜋 over two radians or
90 degrees is zero. This means that when 𝑠 equals
one-half, the work done equals zero. When 𝑠 is equal to zero, we have
negative one over 𝜋 multiplied by the cos of zero. As the cos of zero is one, we are
left with negative one over 𝜋.
The work done between our limits is
therefore equal to zero minus negative one over 𝜋. This simplifies to one over 𝜋. When our force is measured in
newtons and the displacement in meters, then the work done is measured in
newton-meters or joules. We can therefore conclude that the
work done by the force 𝐅 is one over 𝜋 joules.
