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Power & Energy
The rate at which work is done is called as Power.
P is the power at any instant.
F is the force whose power we want to calculate.
V is the velocity at instant when power is being calculated.
- Power can be positive, negative and zero as work done can be positive, negative and zero.
- Power is scaler quantity
- Unit of power is J/s = watt
Q20. A particle of mass m, is projected with speed u at angle theta with horizontal. Find out the power due to gravitation force at time t (t < T, time of flight). Also calculate the average power of gravity force during time t.
We know that at time t= 0, we can take horizontal and vertical component of velocity as shown in figure and can assign them respective unit vector in x direction and y direction (I, j). Also we know that at any time t, velocity in horizontal direction will be same while vertical direction will change according to velocity acceleration equation.
Now, calculate average power at any moment t,
Relation between Power & Kinetic Energy
Assuming velocity of each particle is equal. (In case of rigid body, each particle moves with same velocity)
We can also write this as,
Energy is internal capacity to do work.
Internal capacity to do work by virtue of motion (relative motion) is Kinetic energy.
Internal capacity to do work by virtue of relative position is Potential energy.
Important Notes –
- Potential energy is always defined for a system. We can’t define Potential energy of an isolated object
- Potential energy is always associated with a force and these force are called conservative force
- In case of attraction force, Potential energy increases when separation increases
- In case of repulsive force Potential energy increases when separation decreases
- Potential energy may be positive, Negative or Zero. It depends on the reference point or reference value.
Change in Potential Energy
Change in potential energy is equal to negative of work done by conservative force.
Negative sign shows work done against the nature. As we know that work done on a particle is stored as energy, this energy may be in the form of kinetic energy or potential energy.
Let’s assume there is a system where conservative force is attractive, i.e. particles will not separate. Now if some external force is applied to separate them, work done by this external force will be stored as potential energy in the system. Magnitude of this work done will be equal to that of work done by conservative force to take system into initial position.
Note that, this equation is only applicable when only conservative forces are acting.
Equation of Mechanical energy conservation
This equation is valid when no loss or no supply of energy is there into the system, i.e. No external work has been done on to the system or done by the system.
Mechanical energy is the sum of kinetic energy and all types of potential energies.
ME = KE + PE
- Mechanical energy depends on the reference frame.
- Mechanical energy may be zero, positive or negative
Q21. State TRUE or FALSE
Mechanical energy is always greater than or equal to potential energy, or potential energy can never by greater than mechanical energy.
- By doing external work Mechanical energy of a system can be changed
- Mechanical energy remains unchanged due to work done by internal conservative forces
- Internal resistive (Non-conservative) force can decrease the mechanical energy of system
- If work done by external force and internal resistive forces is zero, we can apply mechanical energy conservation on the system
Consider a particle of mass m & earth as a system. Potential energy is a relative quantity, it depends on the reference point taken.
In case A, potential energy is zero at earth’s surface, hence potential energy of particle can be given as Mgh.
In case B, potential energy at height of 2h from earth is zero, hence potential energy of particle will be –Mgh.
In case C, potential energy at earth’s level is 100, hence particle’s potential energy will be 100 + Mgh.
Change in potential energy is independent of reference frame. Change is same for all observers. So we always take change in potential energy instead of absolute value.
Calculating potential energy of an object
Q22. Calculate potential energy of a rod of length L and mass M. Consider potential energy at surface is zero.
Let’s consider dx length of rod at a height of x from ground. Mass of this dx length of rod is dm. (dx denotes infinitesimal small length)
Potential energy of this dm mass can be written as,
dU = dm.g.x
dm can be written in terms of M, L & dx (assuming density and thickness of rod is uniform)
To calculate potential energy of rod, integrate this equation with respect to x
Q23. Calculate potential energy of a rod which is making an angle theta with surface, as shown in figure below. Assume reference potential energy as zero.
Let’s again right potential energy for dm mass of rod, and integrate it over the length of the rod.
In above example, we’ve learned how to calculate potential energy of an object. For an object potential energy can also be written as
cm is centre of mass of the object.
For the symmetrical objects, centre of mass is the geometrical centre of the object.
So for above case, a circle of mass M & radius R will have potential energy equals to MgR.
Block is released from shown position.
- Find out Mechanical energy of the spring mass system at the shown position.
- Find out M.E. of system when it passes through natural length position
- Find out speed of block at Natural length position
- Max extension in the spring
Mechanical energy = K + U
As there wasn’t any external work done on the system, hence mechanical energy will remain constant in the system. So at natural length position also M.E will remain same.
Now, for speed at N.L. position, apply M.E. conservation equation,
ME at initial position = ME at N.L.
When spring will be at maximum extension position, velocity of block will be zero, apply ME conservation,
Note: Potential energy of spring, , where x is extension or compression in the string from its Natural length position.