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Classical Mechanics
Write down L- lowering operator
Classical Mechanics
determine the tension in each cable in case a and case b use g=10 m/s2
Classical Mechanics
A straight rod of length L is made of a material having mass per unit length m(x)=λ|x| where x is measured from the centre of rod.The moment of inertia about an axis perpendicular to the rod and passing through one end of the rod will be? L=1m and λ=16kg/m^(2)
Ans (32/3) kg/m²
Ans (32/3) kg/m²
Classical Mechanics
simple pendulum consisting of a point mass m on a massless string of length l
attached to the ceiling of a room is oscillating in a plane in the earth gravitational
field.
a) Write down D’Alembert’s equation in cartesian coordinates.
b) Find the constraint of the system and classify it according to holonomic/non-
holonomic and skleronomic/rheonomic.
c) Express the cartesian coordinates of the point mass in appropriate generalized
coordinates. What is the number of degrees of freedom of the point mass?
d) Express the virtual displacements δx and δy in terms of independent general-
ized coordinates.
e) Derive the equations of motion based on D’Alembert’s equation.
f) Solve the equation of motion for small oscillation angles.
attached to the ceiling of a room is oscillating in a plane in the earth gravitational
field.
a) Write down D’Alembert’s equation in cartesian coordinates.
b) Find the constraint of the system and classify it according to holonomic/non-
holonomic and skleronomic/rheonomic.
c) Express the cartesian coordinates of the point mass in appropriate generalized
coordinates. What is the number of degrees of freedom of the point mass?
d) Express the virtual displacements δx and δy in terms of independent general-
ized coordinates.
e) Derive the equations of motion based on D’Alembert’s equation.
f) Solve the equation of motion for small oscillation angles.
Classical Mechanics
Solve the equations of motion for the roll pendulum (cp. Example 3 in the
lectures) for small oscillation angle ϕ 1 and initial conditions:
x1(0) = ˙x1(0) = 0 , ϕ(0) = ϕ0 , ϕ˙(0) = 0 .For small angles ϕ 1, use the approximations sin(ϕ) ≈ ϕ, cos ϕ ≈ 1 and ˙ϕ
2
sin ϕ ≈0
lectures) for small oscillation angle ϕ 1 and initial conditions:
x1(0) = ˙x1(0) = 0 , ϕ(0) = ϕ0 , ϕ˙(0) = 0 .For small angles ϕ 1, use the approximations sin(ϕ) ≈ ϕ, cos ϕ ≈ 1 and ˙ϕ
2
sin ϕ ≈0
Classical Mechanics
Write down equations for the following constraints of mechanical systems
and classify them with respect to whether they are scleronomic or rheonomic and
holonomic or not nonholonomic.
a) A small ball which rolls down on the surface of a big ball without friction.
b) The suspension point of a planar pendulum carries out harmonic horizontal
oscillations.
c) A bead moving freely on a long straight wire through the origin, which rotates
in the x-y plane with constant angular velocity ω about the z-axis.
d) A bead slides along a straight wire. The wire has constant inclination α
towards the horizontal x-y-plane and moves with constant acceleration in x
direction.
and classify them with respect to whether they are scleronomic or rheonomic and
holonomic or not nonholonomic.
a) A small ball which rolls down on the surface of a big ball without friction.
b) The suspension point of a planar pendulum carries out harmonic horizontal
oscillations.
c) A bead moving freely on a long straight wire through the origin, which rotates
in the x-y plane with constant angular velocity ω about the z-axis.
d) A bead slides along a straight wire. The wire has constant inclination α
towards the horizontal x-y-plane and moves with constant acceleration in x
direction.
Classical Mechanics
There has been a proposal to build a train tunnel
underneath the Atlantic Ocean from England to America.
The suggestion is that in the future the trip of 5000 km
could take as little as one hour.
Assume that half the time is spent accelerating uniformly
and the other half is spent decelerating uniformly with the
same magnitude as the acceleration.
(a) Show that the acceleration would be about 2 m s−2. [2]
(b) Calculate the maximum speed. [2]
(c) Calculate the resultant force required to decelerate
the train.
mass of train = 4.5 × 105 kg [2]
underneath the Atlantic Ocean from England to America.
The suggestion is that in the future the trip of 5000 km
could take as little as one hour.
Assume that half the time is spent accelerating uniformly
and the other half is spent decelerating uniformly with the
same magnitude as the acceleration.
(a) Show that the acceleration would be about 2 m s−2. [2]
(b) Calculate the maximum speed. [2]
(c) Calculate the resultant force required to decelerate
the train.
mass of train = 4.5 × 105 kg [2]
Classical Mechanics
A ball is thrown vertically upwards at a speed of 11.0 m s−1.
What is the maximum height it reaches?
What is the maximum height it reaches?
Classical Mechanics
A basketball is thrown with a velocity of 6.0 m s−1 at an angle of
40° to the vertical, towards the hoop.
(a) If the hoop is 0.90 m above the point of release, will the ball
rise high enough to go in the hoop?
(b) If the center of the hoop is 3.00 m away, horizontally, from
the point of release, explain whether or not you believe
this throw will score in the hoop. Support your explanation
with calculations.
40° to the vertical, towards the hoop.
(a) If the hoop is 0.90 m above the point of release, will the ball
rise high enough to go in the hoop?
(b) If the center of the hoop is 3.00 m away, horizontally, from
the point of release, explain whether or not you believe
this throw will score in the hoop. Support your explanation
with calculations.
Classical Mechanics
A boy throws a ball vertically at a velocity of 4.8 m s−1.
(a) How long is it before he catches it again?
(b) What will be the ball’s greatest height above the point
of release?
(a) How long is it before he catches it again?
(b) What will be the ball’s greatest height above the point
of release?
