Forest UCM Homework

Chapt 1

1.10) A particle moves in a circle (center O and radius R) with constant angular velocity [math]\omega[/math] counter-clockwise. The circle lies in the x-y plane and the particle is on the x axis at time t=0. Show the the particle's position is given by

[math] \vec{r}(t) = R \left ( \cos(\omega t) \hat{i} + \sin(\omega t) \hat j \right)[/math]

Find the particle's velocity and acceleration. What are the magnitude and direction of the acceleration? Relate your results to well-known properties of uniform circular motion

1.15.) Show that the definition of the cross product is equivalent to the elementary definition of [math]\vec{r} \times \vec{s}[/math] is perpendicular to both [math]\vec{r}[/math] and [math]\vec{s}[/math], with magnitude rs\sim \theta and direction given by the right-hand rule. For simplicity let [math]\vec{r}[/math] point along the x-axis ans [math]\vec{s}[/math] lie in the x y plane.

1.17.) (a) prove that the vector product [math]\vec{r} \times \vec{s}[/math] is distributive; that is, that [math]\vec{r} \times \left ( \vec{u} + \vec {s} \right ) = \vec{r} \times \vec{u} + \vec{r} \times \vec {s}[/math]. (b) prove the product rule

[math]\frac{d}{dt} \left ( \vec{r} \times \vec {s} \right ) = \vec{r} \times \frac{d \vec {s}}{dt} + \frac{d \vec {r}}{dt} \times \vec{s} [/math]

1.18.)The three vectors \vec a , \vec b , \vec c are the three sides of the triangle ABC with angles \alpha, \beta, \gamma as shown below. (a) Prove that the area of the triangle is given by any one of the these three expressions:

area [math]= \frac{1}{2} \left | \vec{a} \times \vec{b} \right | = \frac{1}{2} \left | \vec{b} \times \vec{c} \right | = \frac{1}{2} \left | \vec{c} \times \vec{a} \right |[/math]

(b) use the equality of the above three expressions to prove the law of sines

\frac{a}{\sin \alpha} =\frac{b}{\sin \beta} =\frac{c}{\sin \gamma}

1.21.) A parallelepiped has one corner at the origin O and the three edges that emanate from O defined by vectors[math] \vec a[/math] , [math]\vec b[/math] , [math]\vec c[/math]. Show that the volume of the parallelepiped is [math]\left | \vec a \cdot \left ( \vec b \times \vec c\right )\right |[/math]

1.24.) Find the general solution to the first-order differantial equation

[math]\frac{d f }{d t} = f[/math] for an unknown function [math]f(t)[/math].

1.35.)A golf ball is hit from ground level with speed v_0 in a direction that is due east and at an angle of \theta above the horizontal. Neglecting air resistance, use Newton's second law to find the position as a finction of tme, using coordiates with x measured east, y nore, and z vertiaclly up. Find the time for the golf ball to return to the ground and how far it travels in that time.

1.37.) A student kicks a frictionless puck with initial speed v_0, so that it slides staight up a plane that is inclined at an angle \theta above the horizontal. (a) Write down Newton's second law for the puck and solve to give its position as a function of time.(b) How long will the puck take to return to its starting point?

Chapter 2

2.5, 2.6, 2.9,2.13, 2.14, 2.15,2.16, 2.18, 2.21, 2.23, 2.25, 2.27, 2.38, 2.39, 2.53,

Chapt 3

3.1,3.2, 3.8, 3.11,3.14, 3.21, 3.25, 3.27, 3.31, 3.23, 3.23, 3.34, 3.35,

Chapt 4

4.2, 4.4, 4.8, 4.17, 4,20, 4.23, 4.25, 4.28, 4.31, 4.35, 4.36, 4.46, 4.48, 4.53?

Chapt 5

5.3, 5.5,

Forest_Ugrad_ClassicalMechanics#Homework_Assignments