Download Angular momentum and PH101:Tutorial

PH101:Tutorial-6
Angular momentum and fixed axis rotation
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Problem 1: A particle of mass m is moving under a force field $ where
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and are two constants and is the velocity of the particle. Show that %
⁄ is a constant of motion . . ; ⁄ 0, where is the angular momentum
of the particle about the origin. Also show that the particle moves on the surface of
right circular cone with opening angle given by cos ⁄ and the symmetry
axis along .
Problem 2: A uniform rigid rod of length L and mass M is pivoted at one end and
swings in a vertical plane as shown in Fig
Fig.1.. Initially the rod was at rest at a small
angle from the vertical. Suppose a paperweight of mass M/3
/3 is glued at an arbitrary
position on the bar, say at a distance x from the pivot point (neglect the dimension of
the paperweight and the contact forces). Obtain the angular frequency of the swing of
the rod as function of x. At what location of the paperweight, the angular frequency
will be maximum?
Fig.1
Fig.2
Problem 3: A plank of length 22l and mass M lies on a frictionless plane. A ball of
mass m and speed v0 strikes its end as shown in Fig.2.. Find the final velocity of th
the
ball, vf, assuming that mechanical
nical energy is conserved and that vf is along the original
line of motion.
Problem 4: A heavy uniform bar of mass M rests on top of two identical rollers which
are continuously turned rapidly in opposite directions, as shown Fig.3. The centers of
the rollers are a distance 2l apart. The coefficient offriction between the bar and the
roller surfaces is µ. Initially the bar is held at rest with its center at distance xo from
the midpoint of the rollers. At time t=0 it is released. Find the subsequent
quent motion of
the bar.
R
F
b
Fig.3
θ
Fig.4
Problem 5: A Yo-Yo of mass M has an axle of radius b and a spool of radius R. Its
moment of inertia can be taken to be MR2/2. The Yo-Yo is placed upright on a table
and the string is pulled with a force F making an angle with the horizontal as shown
in Fig.4. The coefficient of friction between the Yo-Yo and the table is µ. What is the
maximum value of F for which the Yo-Yo will roll without slipping?
Problem 6: A cone of height h and base radius R is free to rotate about afixed vertical
axis as shown in Fig.5. It has a thin groove cut in the surface. The cone is set rotating
freely with angular speed ωo, and a small block of mass m is released in the top of the
frictionless groove and allowed to slide under gravity. Assume that the block stays in
the groove. Take the momentof inertia of the cone about the vertical axis to be I0. (a)
What is the angular velocity of the cone when the block reaches the bottom? (b) Find
the speed of the block in inertial space (Lab frame) when it reaches the bottom.
Fig.5
Fig.6
Problem 7: A ball with radius aand moment of inertia I = (2/5)ma2 rolling with speed
v0 without slipping on the ground collides inelastically with a step of height h<a as
shown in the Fig.6. Find the minimum velocity for which the ball will tripup over the
step. Assume that no slipping occurs also at the impact point.
Problem 8: A ball with radius R and I = (2/5)mR2 is thrown through the air. It spins
around the axis perpendicular to the (vertical) plane of the motion. Call this the x-y
plane. The ball bounces off a floor without slipping during the time of contact.
Assume that the collision is elastic, and that the magnitude of the vertical component
of velocity vy is the same before and after the bounce. Obtain v’x and ω’ after the
bounce in terms of vx and ω before the bounce.