Download End-semester Examination 2013 Mechanics (PHY102A/N

End-semester Examination 2013!
Mechanics (PHY102A/N- KB/MKV)!
19th November 2013!
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PART A!
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Total time: 1 hours
Total Marks: 40
Name
Roll No.
Section
PART A!
1-5 (30 marks)
6 (10 marks)
!
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TOTAL: !
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INSTRUCTIONS FOR PART A!
1.Please answer all the questions in the space provided below the questions. We will
grade only this portion. Rough work can be done in the sheets given at the back. !
2. Please write brief answers only. They should be within the box.!
3. Your tick [√] must be unambiguous.!
4.Formulas are given in page 2 of Part B. Rough work sheets too are given in Part B.!
5. For q multiple choice questions, more than one choice could be correct. (a) any
incorrect tick will result in zero mark for that particular question; (b) No credit will be given
if you do not tick all the correct options.
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PART A !
!
1. [4 Marks] Compute the curl of a vector field ! F = −yx̂ + xŷ
Solution:!
!
!
(∇ × F) = (∂ x Fy − ∂ y Fx )ẑ = 2 ẑ
2. [5 Marks] !A rod of length l and mass M is hinged to the ceiling. When the rod is
displaced slightly from the equilibrium position, it performs a simple harmonic motion.
Compute the angular frequency of the rod. !
Solution: Torque about the hinge O: τ = Mg(l/2) sinθ.!
!
Αngular momentum about O = !
!
Iθ! =
1 2!
Ml θ
3
!
dL/dt = τ implies that for small θ!
3g
θ!! = − θ
2l
!
! !
!
Ηence, the rod would perform SHM with frequency !
!
3g / (2l)
3. [ 5 Marks] !Potential of an electric dipole of magnitude p is given by U(r,θ) = p cos(θ)/
r2. The direction of dipole is along the z axis. Compute the electric field generated by the
dipole. !
!
!
!
!
!
!
!
!
!
∂ p cosθ ˆ 1 ∂ p cosθ
+θ
∂r r 2
r ∂θ r 2
2 p cosθ
psin θ ˆ
=−
r̂
−
θ
r3
r3
2 p cosθ
psin θ ˆ
E = −∇U =
r̂
+
θ
r3
r3
∇U = r̂
[5 marks] Sketch both equipotential surfaces and the electric field in the following figure. !
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2
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4. [6 Marks] ! A solid sphere of mass M and radius R is connected to a vertical axis
through a frictionless hinge as shown in the figure. Size of the hinge can be ignored. The
spheres rolls without slipping about the vertical axis with angular frequency Ω.!
!
(a) Compute the net angular velocity of the sphere. !
(b) Compute the angular momentum of the sphere wrt the hinge. z
Solution: Rolling without slipping => Ω R = -ΩS R => Ω = -ΩS
!
Therefore, the angular velocity .
R
!
Ω = Ωẑ − Ωx̂
!
x
Angular momentum about the hinge: L = LCM + LaboutCM
!
!
!
!
!
2
2
MR 2 Ωẑ − MR 2 Ωx̂
5
5
7
2
= MR 2 Ωẑ − MR 2 Ωx̂
5
5
L = MR 2 Ωẑ +
5. [5 Marks] ! Two charged particles of magnitude +q and masses m approach each other.
At infinity, the velocity of each particle is v0. Treat v0 ≪ c. The shortest distance between
them will be!
!
3
Solution: Shortest distance = r0 ; !
!
Final energy = Potential energy ! U(r0 ) =
!
q2
4πε 0 r0
Initial energy = kinetic energy = 2x½ m v02!
!
Energy conservation => !
!
!
!
!
!
r0 =
q2
4π mε 0 v02
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Multiple choice questions!
1. Imagine an isolated system with two particles interacting with each other via a central
force. For this system, which of the following quantities are conserved. !
(a) √ Total energy!
(b) Total kinetic energy !
(c) √ Total angular momentum !
(d) √ Total linear momentum. !
2. Which of the following statements are correct?!
(a) Newton’s laws of motion are valid for all speeds.!
(b) Einstein’s theory of relativity is valid for high speeds, but not for small speeds.!
(c) √ According to Einstein’s theory of relativity, no particle can have velocity
greater than speed of light.!
(d) √ Time interval between two events remains invariant under Galilean
transformation.!
3. Which of the following statements are correct.!
(a) The formula F = dP/dt (F is the real force, and P is the linear momentum) is valid in
all reference frames, be it inertial or noninertial.!
(b) √ Motion of a rigid body can be completely described by six variables.!
(c) The total kinetic energy of a rigid body is equal to the kinetic energy of its center of
mass. !
(d) Angular momentum is always in the direction of angular velocity.!
4. An electron is moving with velocity c 3 / 2 , where c is the speed of light. Its kinetic
energy would be approximately !
(a) √ 0.5 MeV (mega electron volt).!
(b) 1.0 MeV!
(c) 1.5 MeV!
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(d) 2.0 MeV!
5. Suppose
F = ρρ̂ + ρφ̂ .
The value of the integral ∲F.dr over a circle of unit radius in
xy plane with origin as the centre is!
(a) 0!
(b) π!
(c) √ 2π!
(d) 4π
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