1. Dynamics and Relative Velocity Lecture 07 l Problem Solving: Dynamics Examples l Relative Velocity

2. Dynamics Problem Solving l Draw a Free Body Diagram (FBD) for each Object l Write down  F = ma for each Object in each Direction l Solve for the Unknown(s)

3. Dynamics Example 1 l Two boxes of mass 5 kg and 10 kg are pushed across a floor (with coefficient of friction of  = 0.2) by a force of 50 N. What is the force each block exerts on the other and the acceleration of each block? y x Box 1 FBD: FNFN FPFP FgFg FfFf F 21 Box 2 FBD: FNFN F 12 FgFg FfFf

4. Dynamics Example 1 l Two boxes of mass 5 kg and 10 kg are pushed across a floor (with coefficient of friction of  = 0.2) by a force of 60 N. What is the force each block exerts on the other and the acceleration of each block? y x Box 1  F=ma F P – F f – F = m 1 a 1 x-direction: F N – F g = 0 y-direction: F – F f = m 2 a 2 x-direction: F N – F g = 0 y-direction: Box 2  F=ma

5. Dynamics Example 1 l Two boxes of mass 5 kg and 10 kg are pushed across a floor (with coefficient of friction of  = 0.2) by a force of 60 N. What is the force each block exerts on the other and the acceleration of each block? Box 1 F P – F f – F = m 1 a F P –  F N – F = m 1 a F P –  m 1 g – F = m 1 a Box 2 F – F f = m 2 a F –  F N = m 2 a F –  m 2 g = m 2 a

6. Dynamics Example 1 l Two boxes of mass 5 kg and 10 kg are pushed across a floor (with coefficient of friction of  = 0.2) by a force of 50 N. What is the force each block exerts on the other and the acceleration of each block?  Solve each equation for acceleration and set them equal:  Solve for F (the force each block exerts on the other):  Solve for a (the acceleration of the blocks): 33.3 N 1.4 m/s 2

7. Dynamics Example 2 l A box of mass 3 kg is pulled on a smooth (frictionless) surface by a second block of mass 2 kg hanging over a pulley. What is the acceleration of each block and tension in the string connecting them? y x Box 1 FBD: FNFN FTFT FgFg Box 2 FBD: FTFT FgFg note: the tension is the same everywhere in the string

8. Dynamics Example 2 l A box of mass 3 kg is pulled on a smooth (frictionless) surface by a second block of mass 2 kg hanging over a pulley. What is the acceleration of each block and tension in the string connecting them? y x Box 1  F=ma F T = m 1 a 1 x-direction: F N – F g = 0 y-direction: F T – F g = m 2 a 2 F T – m 2 g = m 2 a 2 y-direction: Box 2  F=ma

9. Dynamics Example 2 l A box of mass 3 kg is pulled on a smooth (frictionless) surface by a second block of mass 2 kg hanging over a pulley. What is the acceleration of each block and tension in the string connecting them?  Substitute the expression for tension from Box 1 into the Box 2 equation:  Solve for a (the acceleration of the blocks):  Solve for F T (the the tension in the string): 11.76 N 3.92 m/s 2

10. Relative Velocity l Sometimes your velocity is known relative to a reference frame that is moving relative to the earth. è Example 1: a boat moving relative to water, which is then moving relative to the ground. è Example 2: a plane moving relative to air, which is then moving relative to the ground. l These velocities are related by vector addition: »v ac is the velocity of the object relative to the ground »v ab is the velocity of the object relative to a moving reference frame »v bc is the velocity of the moving reference frame relative to the ground

11. Relative Motion Example l You are in a plane traveling (relative to still air) at 200 mph at an angle of 30º east of north. There is a wind blowing 50 mph due east (relative to the ground). What is the velocity of the plane with respect to the ground? 30º 50 mph 200 mph y x

12. Relative Motion Example l You are in a plane traveling (relative to still air) at 200 mph at an angle of 30º east of north. There is a wind blowing 50 mph due east (relative to the ground). What is the velocity of the plane with respect to the ground? è To find the velocity with respect to the ground we just need to add the velocity of the plane with respect to the air to the velocity of the air with respect to the ground: v pa v pg v ag

13. Relative Motion Example l You are in a plane traveling (relative to still air) at 200 mph at an angle of 30º east of north. There is a wind blowing 50 mph due east (relative to the ground). What is the velocity of the plane with respect to the ground? è Break the plane’s velocity (with respect to the air) into components. è Add the respective components together to attain the x- and y- components of the net velocity we are looking for! x-direction: v pgx = v pax + v agx v pgx = (200 mph)sin30º + (50 mph) v pgx = 150 mph y-direction: v pgy = v pay + v agy v pgy = (200 mph)cos30º + (0 mph) v pgy = 173 mph

14. Relative Motion Example l You are in a plane traveling (relative to still air) at 200 mph at an angle of 30º east of north. There is a wind blowing 50 mph due east (relative to the ground). What is the velocity of the plane with respect to the ground? è With the components of the net velocity known, find the magnitude and direction of the velocity! v pg 173 mph 150 mph V pg = 229 mph  = 41º 

15. Summary of Concepts l X and Y directions are Independent! l  F = ma applies in both x and y direction l Relative Motion (add vector components)

16. 30º 50 mph 200 mph v pa v pg v ag v pg 173 mph 50 mph