{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

final_review_fri_soln

# final_review_fri_soln - PHY 7A Review for Final...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: PHY 7A - Review for Final Exam (Solutions) Dimitri Dounas-Frazer Spring 2008 Strategy for Harmonic Motion Problems 1. Draw a picture. Choose an origin and a define a set of coordinate axes. Make sure to draw the system slightly displaced from equilibrium. If necessary, choose a pivot point as well. 2. Draw an extended free body diagram (FBD). Decompose all forces along the axes. 3. Apply ∑ F i = m a and/or ∑ τ i = I α . Make approximations if necessary (e.g., sin θ ≈ θ ). 4. Solve for the equation of motion of the system: d 2 /dt 2 =- ω 2 . Identify the angular frequency ω in terms of the parameters of the problem. 5. Solve. Check your answer (units, limits, etc.). Problem 1 1. Draw a picture. Choose an origin and coordinate axes. There are three forces acting on the ball: gravity F , friction f , and the normal force N . Because the ball rolls back and forth along the tunnel, a convenient set of axes is parallel ( x ) and perpendicular ( y ) to the tunnel. With this convention, f points in the x-direction and N points in the positive y-direction. The gravitational force F points from the center the ball to the center of the earth. Choose the origin O at the earth’s center of mass and the pivot point P at the ball’s center of mass. 2. Draw an extended FBD. Decompose forces along axes. Because f and N already point along the axes, we only need to decompose F . The magnitude F of F is given by F = GM m r 2 = GMmr R 3 = mg R r, (1-1) where r is the distance from the center of the earth to the center of the ball. Here m is the mass of the ball, M = M ( r 3 /R 3 ) is the mass of sphere “underneath” the ball, and M and R are the mass and radius of the earth, respectively. (See pages 142-143 of Giancoli.) I have used g = GM/R 2 = 9 . 8 m / s 2 to simplify this expression. Let x and y denote the horizontal and vertical distances from the origin to the center of the ball. After drawing a right triangle, you can see that F x =- mg R x and F y =- mg R y, (1-2) where F x and F y are the components of F along the x- and y- axes, that is, F = F x ˆ i + F x ˆ j . 1 3. Apply Newton’s second law. Since there is no motion in the y-direction, we ignore it com- pletely. In the x-direction, we have X F x,i =- F x + f = ma, (1-3) where a is the acceleration of the ball’s center of mass. Furthermore, the torque equation is X τ i = r obj f...
View Full Document

{[ snackBarMessage ]}

### Page1 / 5

final_review_fri_soln - PHY 7A Review for Final...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online