PHYS 3032 (Spring 2015) Tutorial 7 Classwork
1. The Coriolis force can produce a torque on a spinning object. To illustrate this, consider a
horizontal hoop of mass m and radius r spinning with angular velocity about its vertical
axis at colatitude . Show
Chapter 9
Motion of Rigid Bodies in 3D
Angular Momentum for an Arbitrary Angular Velocity
v i mi ri ( ri )
The angular momentum L mi ri =
for an arbitrary axis passing through a fixed point
An arbitrary angular velocity = ( x , y , z ).
A double cross-
PHYS 3032 (Spring 2015)
Homework 10 (20 points each)
Due time: 10:30am on May 4, 2015
1. A paraboloid is bounded by z = (x2 + y2)/b and the plane z = b.
(a) Find the coordinates of its centre of mass.
(b) Find its moment of inertia about its symmetry axis
Comments on HW10
Question 1, surprisingly many students got integration wrong. Also, check your
units.
Question 2, well done.
Question 3, well done.
Question 4, some students did not, or could not correctly find an approximation in
terms of small ratio m/
PHYS 3032 (Spring 2015)
Homework 9 [20 points each]
Due time: 11:50 m on April 27, 2015
1. Use the method of Lagrange multipliers to find the tensions in the two strings of the double
Atwood machine of Example 10.5.4 in the textbook, i.e. the example in t
PHYS 3032 (Spring 2015)
Homework Solution 6
Problem 1:
(a)
Circular motion of radius
(b)
(c)
where
Let
Try a solution of the form
So
Also at the coordinate systems coincide so
so,
Thus,
Problem 2:
Let the rotating non-inertial frame be denoted by the prim
PHYS 3032 (Spring 2015)
Homework 8 (Total 100 points)
Due time: 10:30am on 20 April 2015
1. (20 points) A particle slides on a smooth inclined plane whose inclination is increasing at
a constant rate . If at , at which time the particle starts from rest ,
Comments on HW8
Question 1, not required.
Question 2, students could find the differential equation, but some could not show
that particle remains between two horizontal circles.
Question 3, it seems that many students still dont understand calculus of va
PHYS 3032 (Spring 2015)
Homework 6 (20 points each)
Due time: 10:30am on March 30, 2015
1.
2. A cockroach crawls with constant speed in a circular path of radius b on a phonograph
turntable rotating with constant angular speed . The circular path is conce
Homework 5 Solution
Problem 1
We are given the satellites heights at perigee and at apogee. The corresponding distances from
the earths center are and . We know from (8.50) that and ,from which it follows that
From this we can find This is the distance fr
PHYS 3032 (Spring 2015)
Homework 7 [20 points each]
Due time: 10:30 am on April 13, 2015
1. Find the differential equations of motion for an elastic pendulum: a particle of mass m
attached to an elastic string of stiffness K and unstretched length l0. Ass
PHYS 3032 (Spring 2015)
Homework 3
Due time: 10:30am on 2 March, 2015
Problem 1 (a, d only) (15 points)
Problem 2 (a, d only) (15 points)
Problem 3 (20 points)
Problem 4 (25 points)
Problem 5 (25 points)
A bee goes out from its hive in a spiral path given
Homework 4 Solution
Problem 6.3, 6.4 and 6.7:
6.3
The gravitational force on the particle is due only to the mass of the earth that is inside
the particles instantaneous displacement from the center of the earth, r. The net effect of
the mass of the earth
PHYS 3032 (Spring 2015)
Homework Solution 7
Problem 1:
,
Problem 2:
(a) For x the distance of the hanging block below the edge of the table:
and
,
(b)and
,
Problem 3:
Let x be the slant height of the particle
,
The solution to the homogeneous equation
,
PHYS 3032 (Spring 2015)
Homework Solution 9
Problem 1:
Note: 4 objects move their coordinates are labelled: The coordinate of the movable, massless
pulley is labelled.
Two equations of constraint:
Thus: (1)
(2)
(3)
Now apply Lagranges equations to the mov
Chapter 7
Dynamics of Systems of N Particles
The Center of Mass
The motion of the Centre of
Mass is a simple parabola.
(just like a point particle)
The motion of the entire
object is complicated =
(motion of the CM) +
(motion of points
around the CM)
vide
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PHYS 3032 (Spring 2015) Quiz 3, 27 April 2015
Duration: 20 minutes (10 points)
A block of mass M is rigidly connected to a massless circular track of radius a on a frictionless
horizontal table as shown below. A particle of mass m is confine
PHYS 3032 (Spring 2015) Quiz 1, 2 March 2015
Suggested Solutions
Problem 1 (20 Points)
Setting and plot to get a sense of how V changes with x over the domain , we have the
following diagram:
We can observe that there is a potential well somewhere in and
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PHYS 3032 (Spring 2015) Quiz 2, 16 March 2015
Duration: 20 minutes (10 points)
Particles of mud are thrown from the rim of a rolling wheel. If the forward speed of the
wheel is , and the radius of the wheel is , show that the greatest height
PHYS 3032 (Spring 2015) Quiz 2, 16 March 2015
Suggested solutions
The maximum height for a projectile motion is where is the initial upward speed.
The initial position of the mud is . Hence the maximum height of the mud is
For the optimal angle, we have
S
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PHYS 3032 (Spring 2015) Quiz 1, 2 March 2015
Duration: 20 minutes (10 points)
(a) A particle of mass m moves under a conservative force with potential energy
where c and a are positive constants. Find the position of stable equilibrium and t
PHYS 3032 (Spring 2015) Quiz 3, 27 April 2015
Suggested Solutions
(a) As the motion of the system is confined to a vertical plane, use a fixed coordinate
frame x, y and choose the x coordinate of the centre of the circular track and the angle
giving the
Comments on Quiz 2
Most students could write the expression for height h in terms of time and angle,
.and knew how to find time at which particles will reach greatest height
Students should remember that height is a function of angle and time, h(t, ), so
Comments on Quiz1
For part (a), many students do not know how to show the equilibrium point is
stable/unstable. Some algebraic mistakes are found when differentiating the
potential, this part should be carefully done otherwise one may get incorrect
answer
HW8 Suggested solutions
Problem 1. Let x be the slant height of the particle
,
The solution to the homogeneous equation
, is
Assuming a particular solution to have the form ,
At time , and
From Appendix B, we use the identities for hyperbolic sine and co
PHYS 3032 (Spring 2015)
Homework Solution 11
9.1 (a)
and
From the perpendicular axis theorem:
(b)
,
,
From equation 9.1.10
(c)
From equation 9.1.29
(d) From equation 9.1.32:
9.3 (a)
From equation 9.2.13,
From Prob. 9.1, , ,
The 1-axis makes an angle of
PHYS 3032 (Spring 2015)
Homework Solution 10
Problem 1:
a) The centre of mass is on the z-axis.
Coordinates: (0, 0, 2b/3).
b)
c)
Problem 2:
The period of the seconds pendulum is where l = 1 m.
The period of the modified pendulum is seconds where, , , n =