Physics - IDepartment of Physics

Indian Institute of Technology Guwahati

Course: PH 101 Total Marks: 30

Date: 23/09/2011 Time: 2-4.00 pmMid-Semester Examination

Marks will be deducted for illegible presentation.

1. (a) A rod of uniform cross-sectional area is maintained vertically on a table. The

linear density of the rod varies as ρ(z) = ρ0 z/l, where l is the length of the rod

and ρ0 is a constant. Gravity acts along negative z-direction. Find the potential

energy of the rod with respect to the table, in terms of ρ0, l, g. [3]

g z

(b) A particle of mass m slides without friction on the inside of a cone (see figure).

The axis of the cone is vertical, and gravity acts downward. The apex half-angle

of the cone is θ. The path of the particle happens to be a circle in a horizontal

plane. The speed of the particle is v0. Find the radius of the circular path in

terms of v0, g and θ. [3]

g

2. A block of mass m slides on a friction less table. It is constrained to move along the

inner surface of a ring of radius R, which is fixed on the table. At t = 0 the block has

a velocity v0 as shown in the figure. The coefficient of friction between the block and

the ring is µ.

i. Find the angular frequency, ω(t), of the block as a function of time. [4]

ii. Find the time taken by the block to make one full revolution of the ring (starting

from t = 0). [2]

R

PleaseTurnOver . . .

1

.

3. A block of mass M slides along a horizontal table. From x = 0 the mass starts

experiencing forces due to a spring and the frictional force of the table (see figure).

The spring has a spring-constant k. The coefficient of friction in this case is

a function of x and is given by µ = bx (for x > 0), where b is a constant. The

spring and board B are massless. At x = 0 the block has a speed v0. Find the loss

of mechanical energy of the block during its motion from x = 0 to the point where it

comes momentarily to rest, for the first time. [5]

B

4. (a) A block moves along the x-axis on a horizontal friction less table. The table is

covered with dust of linear density ρ. As the block moves it gathers all the dust

in the path and gets heavier. At t = 0, the block’s velocity is v0 and its mass is

M0.

i. Find the velocity of the block as a function of distance, x. [3]

ii. What horizontal force, F (x), must be applied on the block, if the block has

to move with constant velocity v0. [2]

(b) A rubber ball of mass m dropped from a height H under gravity hits the ground

and rebounds to a height h. Find the impulse it received from the ground. [2]

5. A bead of mass m slides without friction on a smooth rod along the X-axis. Two

particles each of mass M are located at (0, +a) and (0, −a) as shown in the figure.

The bead is under the gravitational attraction of the masses, M .

i. Starting from the expression of the total potential energy obtain the force on the

bead along the x-direction. [2]

ii. Find the frequency of small oscillation of the bead about the equilibrium. [4]

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