Indian Institute of Information Technology Vadodara
PH101: Physics-I
#Tutorial -2
1. Expresstheunitvectorsrˆ,θˆ,φˆ intermsofxˆ,yˆ,zˆ.Checkyouranswers
several ways (rˆ · rˆ = 1, ˆ θ · φˆ = 0, rˆ × θˆ = φˆ). Also work out the
inverse formulas, givingxˆ, yˆ, zˆin terms ofrˆ, θˆ, φˆ(andθ,φ).
2. Check the divergence theorem for the function v1 = r2r ˆ, using as your
volume the sphere of radiusR, centered at the origin. (b) Do the same for
v2= (1/r2) rˆ.
3. Express the cylindrical unit vectors sˆ, φˆ, zˆ in terms ofxˆ,yˆ,zˆ.“Invert”
your formulas to getxˆ, yˆ, zˆin terms ofsˆ, φˆ, zˆ(andφ).
4. A particle moves outward along a spiral. Its trajectory is givenbyr=A𝛩,
whereAisaconstant.A=(l/𝞹)m/rad.𝛩increasesintimeaccordingto(𝛩=
𝞪 (t2/ 2), where 𝞪 is a constant.
a . Sketch the motion, and indicate the approximate velocity and accel·
eration at a few points.
b. Show that the radial acceleration is zero when𝛩 = 1/ 2rad.
c. At what angles do the radial and tangential accelerations have equal
magnitude?
5. Three freight cars of mass M are pulled with force F by a locomotive.
Friction is negligible. Find the forces on each car.
6. A bead slides without a frictiononarigidwirerotatingatconstantangular
speed ꞷ. Find the force exerted by the wire on the bead (Assume wire is
frictionless and neglect gravity).
7. Is Earth an inertial frame?
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