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.
Work done by a force acting at acute angles to displacement of a system is +ve.
Work done by a force acting at obtuse angles to displacement of a system is – ve.
+ve work increase the KE of the system.
– ve work decrease the KE of the system.
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The y component of the gravitational pull mg cosθ presses the plane & is balanced by normal reaction from the plane, thus N = mg cosθ.
The x component of the gravitational pull mg sinθ (say, F) tends to move the block down the plane, but can do so, only if it exceeds the limiting friction (fL = msmg cosθ) on the block.
As the inclination θ is increased mgsinθ increases while the limiting frictional force (mS mg cosθ) decreases. At one stage, mgsinθ & mS mg cosθ become equal & balance each other then the body placed of the inclined plane at the verge of motion. If the block at this stage is given some velocity it will keep sliding down at constant velocity.
This angle θ is called angle of repose & abbreviated by
symbol β. At θ = β,
or mg sin θ = mS mg cosθ
or tanβ = μS
Also we can make following conclusions
If θ < β, then F < fL & block remains at rest. Friction on block is static & equals mg sinθ.
If θ = β, then F = fL & block remains at rest. Friction on block is static & equals mg sinθ or msmg cosθ.
If θ > β, then friction acting on the block is kinetic mK mg cosθ. This is less than mg sinθ & the block slides down the incline with a uniform acceleration. Using kinematic equations for UAM it can be checked that a block starting from rest will strike the ground with speed in time This is independent of mass of block.
If the inclined plane is smooth (i.e. μ = 0), then the acceleration of a body sliding down a of inclination is, a = g sin θ
Retardation by friction on a rough horizontal surface (i.e. θ = 00) is, a = μ g
The force required to move the block up along the inclined plane with const. acceleration a is, F = mgsinθ + μmgcosθ + ma
/wp-content/uploads/2021/04/logo-2.png00kp-web-admin/wp-content/uploads/2021/04/logo-2.pngkp-web-admin2022-01-17 19:10:052022-01-17 19:10:05^Motion on a rough inclined plane
When seen from inertial frame two types of forces act is a circular motion, one that changes speed & the other that changes direction.
A force that acts tangential of velocity changes the speed only, called tangential force. Its
magnitude is given by
A force acting normal to velocity towards turning centre, along the radius & changes direction is called centripetal force or radial force.
It is not a different type of force. It is actually the resultant of the forces acting on a system & directed towards the center of the circle. For planet revolving around sun gravitational force is centripetal. For a string whirled in a horizontal circle tension in the string is centripetal force. For oscillating pendulum resultant of tension in the string & normal component of weight is centripetal.
The electrostatic force of attraction between electrons & nucleus changes the direction of electrons revolving around the nucleus thus we can write
FCP = Felctrostatic
Gravitational force of attraction between the moon & the earth changes the direction of moon thus we can
write, FCP = Fgravitational
Magnitude of centripetal force on a mass moving with velocity v in a circular path of radius r at constant speed (i.e. uniform circular motion) is
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Basic building blocks of digital electronics made of semiconducting material & used to control the flow of information from input to output in a logical manner & are used in calculators, digital watches, computers, robots, industrial control systems, and in telecommunications.
Transconductance is also called mutual conductance.
As β > > α, thus the ac power gain of a CEA is much larger than that of a CBA. Remember the transistor is not generating any power. The energy for the higher ac power at the output is supplied by the battery.
Both the junction are forward biased & here IC is maximum & does not depend on the input current IB.
Cut off state: Both the junctions are reverse biased as a result IC = 0. Between cut off & saturation state a transistor works as switch as here it turns over rapidly from OFF state (i.e. IC = 0 or cut off) to the ON state (i.e. IC is maximum or saturation state).
Active state: Emitter base junction is forward biased and the collector base junction is reverse biased. A transistor works as an audio amplifier in this regions.
Relation between α and β:
As the value of IB is about 1 – 5 % of IE or IC is 95 – 99 % of IE, α is about 0.95 and 0.99 and β is about 20 to 100. It is found that α and β are independent of current if the emitter base junction is forward biased and the collector base junction is reverse biased. Also the above definitions of α and β do not hold when both the junctions of a transistor are forward biased or reverse biased.
The CE configuration is frequently used as it gives high current gain as well as voltage gain.
Emitter is the section on one side of transistor that supplies charge carriers. It is of moderate size & heavily doped. Collector is the section on the other side of transistor that collects the charge carriers. It is moderately doped but larger in size. Base is the middle section of transistor that forms two pn junctions with emitter and collector. It is very thin and lightly doped so as to pass most of the emitter injected charge carriers to the collector.
A transistor is also called Bipolar Junction Transistor (BJT), as in it current is due to both majority & minority charge carriers.
Input of a transistor is always forward biased, output is reverse biased & the common terminal is grounded.
Current in both npn & pnp transistor is IE = IB + IC. As the base current IB is very small thus IE » IC.. A transistor transfer almost the same electric current from low resistance path to high resistance path and hence is named as transistor. This transistor action makes it useful in transistor amplifier.
A transistor can be used as a switch, amplifier, oscillator & NOT gate.
It is a specially designed junction diode, which can operate in the reverse breakdown voltage region continuously without being damaged. In forward bias behaves like ordinary diode. Zener diode does not gets damaged at breakdown voltage, but it does so at some higher reverse voltage, known as its ‘burn out value’. The magnitude of zener voltage VZ can be decreased by increasing doping level in p and n type materials of zener diode.
I – V characteristics of a Zener diode are as shown. For a voltage more than zener breakdown voltage current through diode increases without increase in voltage across it. This feature is exploited to provide voltage stabilization across a circuit.
Suppose a load RL connected between a & b is connected to a fluctuating dc voltage supply Vs between c & d. Let the load can’t tolerate a voltage VL but the supply Vs can be more than VL. To protect the load from any increase voltage a zener diode of breaking voltage VZ = VL is connected to the input supply with a safety resistor RS in the input loop.
If at any time VZ increases than VL the zener diode works in the breakdown region increasing the current through it without increasing the voltage across it & thus the load always remains protected.