Spring-coupled masses

Here since we are only interested in digits at units place, so to find the digit at units place for a given series we need to carry sum of digits at units place only and will retain the digit at units place eventually.

Further keep in mind that

Digit at units place for 1! = 1

Digit at units place for 2! = 2

Digit at units place for 3! = 6

Digit at units place for 4! = 4

Digit at units place for 5! = 0

Digit at units place for 6! = 0

Digit at units place for 7! = 0,

and will be zero for n!, where n >4

So, a₈= digit at units place of the number (1+2+6+4) = 3

Similarly a₉= 3, .... a₁₆= 3

or a₈ + a₉ + a₁₀ +...+a₁₆= 3 + 3 + ....(nine times) = 27

The total no. of combinations of 2n different things taken any one or more at a time is given by

²ⁿC_1⁺²ⁿC_{2 ⁺}²ⁿC_3⁺²ⁿC_{4 ^{+... +}}²ⁿC_{2n ⁼}2²ⁿ- 1 .......(1)

Similarly, the total no. of combinations of n different things taken any one or more at a time is given by

ⁿC_1⁺ⁿC_{2 ⁺}ⁿC_3⁺ⁿC_{4 ^{+... +}}ⁿC_{n ⁼}2ⁿ- 1 .......(1)

Further it is Given that

(2²ⁿ- 1)/(2ⁿ- 1) = 65/1

let x = 2ⁿ

therefore, (x²- 1)/(x- 1) = 65/1

or x + 1 = 65

or x = 64

or 2ⁿ= 64

or n = 6

I guess the problem is clear and understandable now

HYDRAULIC PRESSURE

pressure applied to any part of a confined fluid transmits to every other part with

no loss. The pressure acts with equal force on all equal areas of the confining walls

and perpendicular to the walls. This is Hydraulic pressure.

GUAGE PRESSURE

Gauge pressure is referenced against ambient air pressure, so it is equal to absolute

pressure minus atmospheric pressure.

In general both are two different things.

ELECTRIC potential energy is due to the presence of charge or electric field in which the sphere is placed.

Whereas total energy of the sphere includes electric potential energy as well as kinetic energy (if any) of the sphere.

Further the difference may arise in context to a particular question.

An electrical generator is a device that converts mechanical energy to electrical energy, generally using electromagnetic induction.

There is a difference of commutator in ac and dc generator.

AC generator is without commutator.

DC generator is with commutator.

Please click on the following link to view animation of ac and dc generator

http://www.sciencejoywagon.com/physicszone/lesson/otherpub/wfendt/generatorengl.htm

The frictional force always opposes the motion of the body.

Keeping this in mind take direction of frictional force to be opposite to the directionof motion of the body. No matter whether it is translational motion or rotational the direction of resistive/frictional force will always be in the opposite direction.

Apply law of conservation of momentum to find the velocity of the bullet when it emerges out.

Now the kinetic energy of block will be such that it acquires potential energy so that the center of mass of the block raises by 20cm

Use (1/2)mv² = mgh, find v for the block.

substitute in the equation for conservation of momentum and get the required result.

In electrical circuit theory, Thevenin's theorem for electrical networks states that any combination of voltage sources, current sources and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem can also be applied to general impedances, not just resistors.

Any black box containing only voltage sources, current sources, and resistors can be converted to a Thévenin equivalent circuit.

Any black box containing only voltage sources, current sources, and resistors can be converted to a Thevenin equivalent

STEPS FOR SOLVING THE NETWORK USING THEVENIN'S THEOREM

Calculate the output voltage, V_AB, when in open circuit condition (no load resistor - meaning infinite resistance). This is V_Th.
Calculate the output current, I_AB, when those leads are short circuited (load resistance is 0). R_Th equals V_Th divided by this I_AB.

The equivalent circuit is a voltage source with voltage V_Th in series with a resistance R_Th.

Case 2 could also be thought of like this:

2a. Now replace voltage sources with short circuits and current sources with open circuits.

2b. Replace the load circuit with an imaginary ohm meter and measure the total resistance, R, "looking back" into the circuit. This is R_Th

Example of a Thévenin equivalent circuit

Step 0: The original circuit

Step 1: Calculating the equivalent output voltage

Step 2: Calculating the equivalent resistance

Step 3: The equivalent circuit

In the example, calculating equivalent voltage:

$V_\mathrm{AB} = {R_2 + R_3 \over (R_2 + R_3) + R_4} \cdot V_\mathrm{1}$

$= {1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega \over (1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega) + 2\,\mathrm{k}\Omega} \cdot 15 \mathrm{V}$

$= {1 \over 2} \cdot 15 \mathrm{V} = 7.5 \mathrm{V}$

Calculating equivalent resistance:

$R_\mathrm{AB} = R_1 + \left ( \left ( R_2 + R_3 \right ) \| R_4 \right )$

$= 1\,\mathrm{k}\Omega + \left ( \left ( 1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega \right ) \| 2\,\mathrm{k}\Omega \right )$

$= 1\,\mathrm{k}\Omega + \left({1 \over ( 1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega )} + {1 \over (2\,\mathrm{k}\Omega ) }\right)^{-1} = 2\,\mathrm{k}\Omega$

Work done will be for displacement 'ds' is F.ds only,

Integration can be performed if limits of x and y are given.

To find the conservative nature of force use following tips:

A force F is called conservative if it meets any of these conditions:

The curl of F is zero:

$\nabla \times \vec{F} = 0. \,$

The work, W, is zero for any simple closed path:

$W = \oint_C \vec{F} \cdot \mathrm{d}\vec r = 0.\,$

The force can be written as the gradient of a potential, $?$ :

$\vec{F} = -\nabla \Phi. \,$

Conservative force fields are curl-less as a direct consequence of Helmholtz decomposition. The term conservative force comes from the fact that when a conservative force exists, it conserves mechanical energy. The most familiar conservative forces are gravity, the electric force, and spring force.

Rolling resistance, sometimes called rolling friction, is the resistance that occurs when an object (e.g a wheel or tire) rolls. It is much smaller than sliding friction except for special cases like ice skating. It is caused by the deformation of the wheel or tire or the deformation of the ground. It depends very much on the material of the wheel or tire and the sort of ground. For example, rubber will give a bigger rolling friction than steel. Also, sand on the ground will give more rolling friction than concrete. A vehicle rolling will gradually slow down due to rolling friction, but a train with steel wheels running on steel rails will roll much further than a car or truck with rubber tires running on pavement, even when differences in mass and momentum are accounted for.

Factors contributing towards rolling friction

Several factors affect the magnitude of rolling friction a tire generates:

Material - Tires with higher sulfur content tend to have a lower rolling friction. This is one strategy that most hybrid car vendors use to improve fuel efficiency.
Dimensions - rolling friction is related to the flex of sidewalls and the contact area of the tire.
Extent of inflation - Lower pressure in tires results in more flexing of sidewalls and higher rolling friction.
Over inflating tires (such a bicycle tires) may not lower the overall rolling resistance as the tire may skip and hop over the road surface. Traction is sacrificed, and overall rolling friction may not be reduced as the wheel rotational speed changes and slippage increases.
Tread thickness has much to do with rolling resistance. The thicker the tread, the higher the rolling resistance. Thus, the "fastest" bicycle tires have very little tread and heavy duty trucks get the best fuel economy as the tire tread wears out.
Smaller wheels, all else being equal, have higher rolling resistance than larger wheels

If mass of moon is already given (point 2) in the option then why do u bother to find it again? Please check your question again.

I think in step 2 it should be mass of earth rather than that of moon.

So if this is the case then to find mass of moon you can use Kepler's Law of planetary motion

T² = 4

²R³/GM

Relation between linear velocity and Angular velocity of a body about any point is given by

= (r x v) / I r I²

where v is linear velocity and r is position vector of the body w.r.t point about which we intend to find angular velocity

Further if the point is on the line of a velocity vector then the angle

is zero so sin

= 0, implying that angular velocity vanishes so

= 0

How can a bullet weigh 20kg?

Just take it on a light note. I know its a typing mistake. Anyway i consider the mass to be 20g for further calculations

Mass of bullet = m₁= 20g = 20/1000 kg = 0.02kg

Mass of block = m₂

Applying law of conservation of momentum after the bullet emerges with velocity 100m/s we get

(0.02)*500 = m₂v + (0.02)*100

or v = 8/m₂

Now block travels a distance of 20cm = 0.2m before coming to rest

using, v²- u²= 2as (put u = 8/m₂ and v = 0)

Now, Find accelertion(-ve) offered by frictional force(F =

mg) . Use frictional force equation

to find the coefficient of friction.

By two body oscillators if you mean coupled oscillations. Then examples are

1) two masses at the two ends of a spring and thus coupled by means of spring

2) In a lattice or solid the neighboring atoms are coupled to each other and thus the motion of one influences the motion of other neighboring atoms.

3) Similarly we talk about coupled harmonic oscillator.

Moreover Coupled Oscillations occur when two or more oscillating systems are connected in such a manner as to allow motion energy to be exchanged between them. Coupled oscillators occur in nature (e.g., the moon and earth orbiting each other) or can be found in man-made devices (such as with the pacemaker).

Coupled oscillations

The harmonic oscillator, and the more complicated systems for which it stands as a simple model, has a single degree of freedom. More complicated systems have more degrees of freedom, for example two masses and two springs. In such cases, energy is converted between the respective inertias of each degree of freedom and the several restoring forces in the system. This leads to a coupling of the oscillations of the individual degrees of freedom. For example, two pendulum clocks mounted on a common wall will tend to synchronise. The apparent motions of the individual oscillations typically appears very

Spring-coupled masses

Consider the two degree of freedom dynamical system pictured in folowing Figure. In this system, two point objects of mass

are free to move in one dimension. Furthermore, the masses are connected together by a spring of spring constant

, and are also each attached to fixed supports via springs of spring constant

.

**Spring Coupled masses**

Let

and

be the displacements of the first and second masses, respectively, from the equilibrium state. If follows that the extensions of the left-hand, middle, and right-hand springs are

,

, and

, respectively. The kinetic energy of the system takes the form

$egin{displaymath} K = rac{m}{2} ,(dot{q}_1^{,2} + dot{q}_2^{,2}), end{displaymath}$

whereas the potential energy is written

$egin{displaymath} U= rac{1}{2}left[k',q_1^{,2} + k,(q_2-q_1)^2+ k',q_2^{,2} ight]. end{displaymath}$

The above expression can be rearranged to give

$egin{displaymath} U= rac{1}{2}left[(k+k'),q_1^{,2} -2,k,q_1,q_2 + (k+k'),q_2^{,2} ight]. end{displaymath}$

Ofcourse it's not late to start aiming and preparing for IIT JEE kind of things at 11th standard. Just keep aiming 2wards the same and strive to excel in the same. take guidance from the experts and start to put ur effort in the same direction. Again i repeat the word hard work, dedication and strong will solve th purpose and shall help u in accomplishing ur desired goal.

All the very best for your future endeavours.