Phys104 W15 Midterm exam solutions
1). (10 points) Prove that the shortest path connecting two points in the x-y plane is a straight line.
This is the Calculus of Variations example that we did repeatedly in class.
We want to nd the function (or path), y(
Homework 2 Solutions
Physics 104
Taylor 7.4
Consider a mass m moving in a frictionless plane that slopes at an angle with the horizontal.
Write down the Lagrangian in terms of coordinates x, measured horizontally across the slope, and
y, measured down the
Homework 2 Solutions
Physics 104
Winter 2015
1 Taylor 7.3
1
Consider a mass m moving in two dimensions with potential energy U (x, y) = 2 kr2 , where r2 = x2 +y 2 .
Write down the Lagrangian, using coordinates x and y, and nd the two Lagrange equations of
Homework 2 Solutions
Physics 104
Winter 2014
1 Taylor 7.50
A mass m1 rests on a frictionless horizontal table. Attached to it is a string which runs horizontally
to the edge of the table, where it passes over a frictionless, small pulley and down to where
Homework 4 Solutions
Physics 104
Winter 2015
1 Taylor 10.11
(a) Use the result of Problem 10.4 to nd the moment of inertia of a uniform solid sphere (mass
M , radius R) for rotation about a diameter. (b) Do the same for a uniform hollow sphere whose
inner
Homework 4 Solutions
Physics 104
Winter 2016
1). Taylor 10.14
A stationary space station can be approximated as a hollow spherical shell of mass 6 tonnes (6000
kg) and inner and outer radii of 5 m and 6 m. To change its orientation, a uniform flywheel (ra
Homework 7 Solutions
Physics 104
Winter 2016
1. Taylor 16.9
The motion of a finite string, fixed at both ends, was determined by the wave equation
(16.19) and the boundary conditions (16.20). We solved these by looking for a solution that
was sinusoidal i
Phys104, W17, Homework 3
Due Thursday, 2/2, by noon in box in Broida lobby
Make sure to fully show your work and explain steps.
To facilitate ordering the graded homeworks and returning them to you, please write your lastname in block letters on the upper
Homework 6 Solutions
Physics 104
Winter 2016
1. Taylor 13.7
A roller coaster of mass m moves along a frictionless track that lies in the x-y plane (x
horizontal and y vertically up). The height of the track above the ground is given by y =
h(x). (a) Using
Phys103 F14 Final exam solutions
You must fully show your work and explain steps to get credit.
See back for a list of possibly useful mathematical relations.
1. (10 points) A sand-spraying locomative sprays sand horizontally into a freight car as shown
i
Phys103 F14 Final exam
(Form A)
You must fully show your work and explain steps to get credit.
See back for a list of possibly useful mathematical relations.
1. (10 points) A sand-spraying locomative sprays sand horizontally into a freight car as shown
in
Homework 3 Solutions
Physics 104
Winter 2016
Taylor 9.1
Be sure you understand why a pendulum in equilibrium hanging in a car that is accelerating forward
tilts backward, and then consider the following: A helium balloon is anchored by a massless string
t
Homework 5 Solutions
Physics 104
Winter 2016
1). Taylor 11.5
(a) Find the normal frequencies, 1 and 2 , for the two carts shown in Figure 11.15, assuming that
m1 = m2 and k1 = k2 . (b) Find and describe the motion for each of the normal modes in turn.
a)
Phys104, W17, Homework 5
Due Monday, 2/27, by 5:00 PM in box in Broida lobby
Make sure to fully show your work and explain steps.
To facilitate ordering the graded homeworks and returning them to you, please write your lastname in block letters on the upp
Homework 1 Solutions
Physics 104
Taylor 6.1
The shortest path between two points on a curved surface, such as the surface of a sphere, is called a
geodesic. To nd a geodesic, one has rst to set up an integral that gives the length of a path on the
surface
Phys 104 Section 6 Evan Losh
Last week, we solved a problem using the Euler equations in the body frame of an object.
We worked in the body frame so that I would be diagonal and not a function of time.
Now we seek to work in a space frame while continuing
Phys 104 Section 6
Gabriel
February 15th, 2017
Euler's Equations
The angular momentum is given by the tensorial equation
(1)
~ = I~
L
where I is the tensor of inertia. We can express
~ in the basis of the principal
axes cfw_e1 , e2 , e3 , then
~ = (1 e1
Homework 4 Solutions
Physics 104
Winter 2017
1). Taylor 10.22
A rigid body comprises 8 equal masses m at the corners of a cube of side a, held together by
massless struts. (a) Use the definitions (10.37) and (10.38) to find the moment of inertia tensor I
Phys 104 Section 5 Evan Losh
When an object rotates about a principle axis (often an axis of symmetry) its moment of
inertia can be treated eectively as a scalar quantity. However, moment of inertia is in
general a 33 matrix, or tensor. It is dened by
Z Z
Homework 5 Solutions
Physics 104
Winter 2016
1). Taylor 11.5
(a) Find the normal frequencies, 1 and 2 , for the two carts shown in Figure 11.15, assuming that
m1 = m2 and k1 = k2 . (b) Find and describe the motion for each of the normal modes in turn.
a)
Phys 104 Section 5 Evan Losh
When an object rotates about a principle axis (often an axis of symmetry) its moment of
inertia can be treated eectively as a scalar quantity. However, moment of inertia is in
general a 33 matrix, or tensor. It is dened by
Z Z
Phys 104 Section 5
Moment of Inertia Tensor
The angular momentum L of a rigid body that is rotating with angular velocity is given by
(1)
L = I
where L and should be seen as 3 1 columns and I is a 3 3 moment of intertia tensor, given by
Ixx =
m (y2 + z2 )
Homework 1 Solutions
Physics 104
Taylor 6.1
The shortest path between two points on a curved surface, such as the surface of a sphere, is called a
geodesic. To find a geodesic, one has first to set up an integral that gives the length of a path on the
sur