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Q1: Two particles move in a uniform gravitational field with an acceleration g. At the initial moment the particles were located at one point.and moved with velocites v1 = 3.0ms-1 and v2 = 4.0ms-1 horizontally in the opposite directions. Find the distance between the particles at the moments when their velocity vector become mutually perpendicular.
Solution 1: Since velocity of the particles are perpendicular to each other, the dot product must vanish. Hence the following
V1⋅V2=0 when v1 ⊥ v2
where we know that i^.i^=1, j^.j^=1 and i^.j^=j^.i^=0 (unit vectors)
Q2: Find the rate v with which helium flows out of a thermally insulated vessel into vacuum through a small hole. The flow rate of the gas inside the vessel is assumed to be negligible under these conditions. The temperature of helium in the vessel is T = 1000K.
Solution 2:From energy conservation as in the derivation of Bernoulli’s theorem it reads
In the Eq.(1) u is the internal energy per unit mass and in this case is the thermal energy per unit mass of the gas. As the gas vessel is thermally insulated, Qd=0, also in our case.
Just inside the vessel u = (CVT/M) = RT/M(γ−1) also p/ρ=RT/M. inside the vessel v=0 also. Just outside p=0, and u=0. In general gz is not very significant for gases.
Thus applying Eq. (1) just inside and outside the hole, we get
(1/2)v2 = p/ρ+u
= RT/M + RT/M(γ−1) = γaRT/M(γ−1)
or, v = √2γRT/M(γ−1)
Q3: A ball of radius R is uniformly charged with the volume density d. Find the flux of the electric field strength vector across the ball’s section formed by the plane located at a distance ro<R from the centre of the ball.
Solution 3: Let us consider a ring element of radius x and thickness dx, Now , flux over the considered element,
dφ = E.dS = ErdScosθ
But, Er = ρr/3ε0 from Gauss’s theorem,
and dS = 2π x dx, cosθ=r0/r
Thus dΦ = (ρr/3ε0) (2π×dx) (r0/r) = ρr02πxdx/3ε0
Hence sought flux
Φ = (2πρr0/3ε0) ∫xdx = (2πρr0/3ε0) [(R2−r02)/2] = (πρr0/3ε0)(R2−r02)
Q4:A plane light wave falls on fresnel mirrors with an angle @ = 2.0′ between them. Determine the wavelength of light if the width of the fringe on the screen delta(x) = 0.55mm.
Solution 4:To get this case we must let r→∞ in the formula for Δx of the lest example.
So Δx = (b+r)λ/2αr → λ/2α.
(A plane wave is like light emitted from a point source at ∞).
Then λ = 2αΔx = 0.64μm.
Q5:A long cylinder of radius R1 is displaced along its axis with a constant velocity vo inside a stationary co-axial cylinder of radius R2. The space between the cylinders is filled with viscous liquid.Find the velocity of the liquid as a function of the distance r from the axis of the cylinders. The flow is laminar.
Solution 5:Let us consider a coaxial cylinder of radius r and thickness dr, then force of friction or viscous force on this elemental layer, F = 2πrlη(dv/dr).
This force must be constant from layer to layer so that steady motion may be possible.
or, F(dr/r) = 2πlηdv (1)
F∫(dr/r) = 2πlη∫dr
or, F log(r/R2) = 2πlηv (2)
Putting r=R1, we get
F log(R1/R2) = 2πlηv0
From (2) by (3) we get,
v = v0 [log(r/R2) / log(R1/R2)]
The force F is supplied by the agency which tries to carry the inner cylinder with velocity v0.
Hope the list of solved questions along with the application of formulas made things easier for you. In case you still need more clarity, why not connect with our online Physics tutor today? Get ready to connect with your inner Einstein by diving deep now. Your tutor will make sure to make lessons engaging and quite fun!
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Memorize the formula:
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Check out its application:
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Score better marks:
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The more you practice them the better you get at arriving at a quick solution.
Let’s Wrap up
Our Grade 11 Physics Formula Sheet is the best tool that will help you prepare for various competitive examinations. To excel in the subject you need to spend time understanding various complex formulas. Your online Physics tutor can come to your rescue and help you to figure out the complicated stuff.
Students are advised to learn these formulas well to excel in the subject. This is where the exciting and interesting journey to learn Physics begins, just be sure that you have the best resource like TutorEye to make it fun.