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]

1.18,1.21,1.24 1.26)

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?

1.26, 1.35, 1.36, 1.37, 1.38, 1.46

1.26,1.27,1.35,1.36,1.37,

Forest_Ugrad_ClassicalMechanics#Homework_Assignments