Block at Rest A 1.5 kg block is initially at rest on a...

Block at Rest A $1.5 \mathrm{~kg}$ block is initially at rest on a horizontal frictionless surface when a horizontal force in the positive direction of an $x$ axis is applied to the block. The force is given by $\vec{F}(x)=$ $\left(2.5 \mathrm{~N}-x^{2} \mathrm{~N} / \mathrm{m}^{2}\right) \mathrm{i}$, where $x$ is in meters and the initial position of the block is $x=0.0 \mathrm{~m} .($ a) What is the kinetic energy of the block as it passes through $x=2.0 \mathrm{~m} ?$ (b) What is the maximum kinetic energy of the block between $x=0.0 \mathrm{~m}$ and $x=2.0 \mathrm{~m}$ ?

Transcript

00:01 Hi guys now in this problem a block at rest which is a mass of 1 .5 kj now we applied a force on this object which is about 2 .5 minus x squared from x is equal to 0 to x is equal to 2 what we need to find out first what is the kinetic energy at x is equal to 2 meter now we apply this thing this force we apply so what we are going to do here in order to find out the kinetic energy at x is equal to 2 that work done is equal to change in kinetic energy work done work done is equal to change in kind dignity initially it has zero kind technology because it is at the rest and when it is at x is equal to 2 let's say it has some velocity that is what be so at x is equal to 2 meter it has some kind technology so k minus 0 is equal to work done now what is work done can be fined as work done is equal to integration of f dot d x so work done is equal to similarly f dot d x that means what the work done this is we need to find out work done from x is equal to 0 to 2 so 0 to 2 and the f is what this is 5 divided by 2 minus x square times d x so this value k is equal to what this is is 5 divided by 2 times x minus x cube divide by 3 let's substitute the limits 0 to 3 substitute 0 you'll get 0 to 2 substituting 2 5 divided by 2 times 2 which is what 5 minus this is 8 divide by 3 2 cube is what 8 8 divide by 3 so this is 7 divide by 3 jule next, what we need to find out? what is the kinetic energy of block? we need to find out the maximum kinetic energy between x is equal to 0 to 2 meter.

02:34 So we already find out the k is what kindic energy between x is equal to 0 to 2, k in the form of x.

02:44 5 divided by 2 times x minus x squared, sorry x squared divided by 3.

02:51 In order to find out the maximum kinetic energy, this is dk divided by d x, we are differentiating this thing with respect to x.

03:00 So 5 divided by 2 minus this is x square is equal to 0.

03:07 So from here you will get two values of x.

03:09 1 is what x is equal to plus minus of, this is a square root of or i can say 5 divide by 2 to the power 1 divided by 2...