Springs

^Springs

1. Spring force does -ve work when we expand or stretch & +ve work when the spring force restores itself from extended or stretched positions.

2.Spring store energy called elastic potential energy whether expanded or stretched from relaxed state.

3.If a block of mass ‘m’ is released on a spring of stiffness constant ‘k’ from a height ‘h’ compresses it by amount  ‘x’ , then: mg (h + x) =

 

Work energy theorem

^Work energy theorem

Total work done by all the forces acting on a system between any two points is equal to change of kinetic energy between those two points.         

Here all forces means real or pseudo, conservative

or non, internal or external.

Conservation of mechanical energy

^Conservation of mechanical energy

K + U = E = constant                                 

or  – (UB – UA) = + (KB – KA)In a conservative field loss of potential energy between any two points is always equal to gain of KE between those two points for any angle between force & velocity.

Work by a conservative force

^Work by a conservative force

Work done by a conservative force in moving a body from one point say ‘A’ to other point say ‘B’  is always equal to

loss of potential energy between same points i.e.,

Here the negative sign implies that a conservative force is always directed in the direction of decreasing

potential energy (i.e. increasing stability).Dropping the integral sign & the dot product in the LCF, we can write,

i.e. a conservative force is equal to the negative of potential energy gradient.

In vector form the above result can be expresses as

For one dimensional force we can write 

 

 

^Potential energy 

^Potential energy 

  1. Potential energy is defined only for conservative forces.
  2. Gravitation, electromagnetic, spring force all are conservative.
  3. Friction is non conservative.
  4. Usually a point at infinity is assigned zero value i.e. U∞ = 0
  5. Potential energy can be + ve, -ve or zero.
  6. Generally –ve potential energy indicates attractive forces & +ve indicates repulsive forces.

^Kinetic energy 

^Kinetic energy 

Kinetic means motion. A mass ‘m’ moving at

As both m & v2 are + ve & scalar, thus the KE of a body is always a +ve scalar quantity. Where as linear momentum is vector and always directed in the direction of velocity.

KE of a system of particles is the sum of kinetic energy of all of its constituent particles. i.e.

KE of a body depends upon the frame of reference. e.g. a person sitting in a moving bus has some KE w.r.t. the person standing at rest on road but no KE w.r.t. the persons in sitting in the same bus.

Actually KE is defined as the amount of work done to accelerate a body from rest.

Proof:       

Rest means both linear momentum & KE zero. A body can’t have KE without linear momentum & vice – versa. Using the relation

we can say that

  1. If the linear momentum of a body is doubled, then its KE becomes 4 times.
  2. If the KE of a body is doubled, then its linear momentum becomes 
  3. If a light body and a heavy body have same linear momentum, then lighter body will have greater KE.
  4. If a light body and a heavy body have same KE, then heavier body will have greater linear momentum.

 

 

^Work to pull a chain

^Work to pull a chain

A chain is held on a frictionless table with 1/n of its length hanging over the edge. If the chain has a length L and a mass M. Work required to pull the hanging part back on the table is, 

 

^Calculating work from F – x graph

^Calculating work from F – x graph

is read as work done by a force ‘Fx’ in moving an object form point A to point B is equal to area under F – x graph bounded with the displacement – axis under position limits of the point A & B.

Conventionally upward areas are +ve & downward –ve.

^Examples of zero,  +ve & -ve work

^Examples of zero,  +ve & -ve work

Zero work means either no displacement of system no net change in the KE of the system. Few examples of zero work are

  1. If there is no motion, no work has been done no matter how much force is applied.
  2. A static person e.g.  a gate keeper does no work, as his displacement is zero.
  3. Work done by normal forces (any force that changes only direction) is always zero e.g. work  done by tension in the sting simple pendulum, by your weight while walking on a horizontal surface, by normal reaction on a body.
  4. Work done by a gas during its free expansion.
  5. Work done by a magnetic field in moving a charge in a circle.
  6. Gravity does positive work on an descending mass, negative work on an ascending mass & zero work for both for horizontal motion & cyclic path.
  7. Consider the case of a weight lifter. He does,
  8. (a) positive work to lift as he lifts weights the weights up (to pull weights up along the displacement against the gravity)
  9. (b) no work, to hold the weights at their position (as no displacement means, no work )
  10. (c) negative work to bring weights down at         constant speed (as he has to pull opposite to displacement to maintain speed)

^Sign of work

^Sign of work

  1. Sign of work depends on sign of cosθ. As cosθ can be 0, +ve or – ve (recall –1 ≤ cosθ ≤ +1), hence the work done by a force also can be 0, + ve or – ve depending upon the angle between the force and the displacement.
  2. Work done by a force acting at acute angles to displacement of a system is +ve.
  3. Work done by a force acting at obtuse angles to displacement of a system is – ve.
  4. +ve work increase the KE of the system.
  5. – ve work decrease the KE of the system.
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