The spring is now compressed so that the unconstrained end moves from x=0 to x=L. Using the work integral W=∫xfxiF⃗ (x⃗ )⋅dx⃗ , find the work done by the spring as it is compressed. Express the work done by the spring in terms of k and L?

solution:

the spring force exerted by spring with spring constant k is given by

where k is spring constant

and x is deformation of spring

in order to calculate word done by the spring

the work done by the spring as it is compressed from x=0 to x=L

inserting the limits x=0 and x=L

we get work done in terms of k and L

air exerts forces on falling objects near earth’s surface.

explanation:

the term 'free falls' refer to object that are falling towards the earth, in a situation in which gravity is the only force acting on the object.

for an object in free fall, the acceleration is constant (g, the gravitational acceleration), and so the velocity of the object constantly increases.

for objects near earth's surface, there is always another force acting on the object: the resistance due to the air. this force is opposed to the direction of fall of the object, and it is proportional to the velocity of the object, so as the object falls down, this force increases up to a point when it balances the force of gravity, and the object continues its motion at constant velocity (called terminal velocity). since in this case gravity is not the only force acting on the object, we are not in a situation of free fall.