Pulleys Example Whiteboards Atwoods Machine Whiteboards Inclined Plane

Pulleys Example Whiteboards - Atwood’s Machine Whiteboards - Inclined Plane M 1 M 2

Example - Find Tension, and acceleration if the force of friction acting on the 32 kg block is 34. 5 N Step 1 - guess direction of + acceleration (This becomes the + direction for each mass) 32. 0 kg Step 2 - Calculate or express <F> = ma for every mass object +a T (-) +a 5. 0 kg +49. 05 N For the 5. 0 kg mass, the tension T is up (-) The weight of 5*9. 81 = 49. 05 N is down (+): And our formula becomes: 49. 05 N - T = (5. 0 kg)a

49. 05 N - T = (5. 0 kg)a Step 1 - guess direction of + acceleration (This becomes the + direction for each mass) -34. 5 N T 32. 0 kg Step 2 - Calculate or express <F> = ma for every mass object The 32. 0 kg mass has the tension T to the right (+) And, assuming it is moving to the right, a frictional force of 34. 5 N to the left (-) +a +a 5. 0 kg So our equation becomes: T - 34. 5 N = (32. 0 kg)a

And now it’s math time!!!! Two equations, two unknowns! 49. 05 N - T = (5. 0 kg)a + T - 34. 5 N = (32. 0 kg)a Add them Together! 49. 05 N - T + T - 34. 5 N = (5. 0 kg)a + (32. 0 kg)a 49. 05 N - 34. 5 N = (37 kg)a a = (49. 05 N - 34. 5 N)/37 kg = 0. 393243243 m/s/s And plug into an equation to find T: T - 34. 5 N = (32. 0 kg)a T = (32. 0 kg)(0. 393243243 m/s/s) + 34. 5 N = 47. 08 N

Whiteboards: Atwood’s Machine 1|2|3|4|5 TOC

Massless frictionless pulley +a 3. 0 kg 5. 0 kg +a Find acceleration and tension Step 1 - Guess the direction of acceleration - This becomes the positive direction for each mass. Uhh well um. the 5. 0 kg is heavier. Green, Bananas never do W

Massless frictionless pulley +a 3. 0 kg 5. 0 kg +a Find acceleration and tension Step 2 - Set up the <F>=ma for the 3. 0 kg mass: T is up , and the weight is down. Down is and up is + weight = (3. 0 kg)(9. 81 N/kg) = 29. 43 N down (- in this case) T is up, -29. 43 is down: T - 29. 43 N = (3. 0 kg)a W

Massless frictionless pulley +a 3. 0 kg 5. 0 kg +a Find acceleration and tension Step 3 - Set up the <F>=ma for the 5. 0 kg mass: T is up , and the weight is down, but now down is + and up is - weight = (5. 0 kg)(9. 81 N/kg) = 49. 05 N down (+ in this case) T is up (-), 49. 05 N is down (+): 49. 05 N - T = (5. 0 kg)a W

Massless frictionless pulley +a 5. 0 kg 3. 0 kg +a Find acceleration and tension Step 4 - Solve for acceleration: 49. 05 N - T = (5. 0 kg)a T - 29. 43 N = (3. 0 kg)a 49. 05 N - T = (5. 0 kg)a +T - 29. 43 N = (3. 0 kg)a T - 29. 43 N + 49. 05 N - T = (8. 0 kg)a a = 2. 4525 m/s/s 2. 45 m/s/s W

Massless frictionless pulley +a 5. 0 kg 3. 0 kg +a Find acceleration and tension Step 5 - Solve for T: 49. 05 N - T = (5. 0 kg)a T - 29. 43 N = (3. 0 kg)a a = 2. 4525 m/s/s Pick one of the formulas, and plug the acceleration in: T - 29. 43 N = (3. 0 kg)a T = (3. 0 kg)(2. 4525 m/s/s) + 29. 43 N = 36. 7875 N ≈ 37 N W

Whiteboards: Pulleys on Inclined Planes 1|2|3|4|5 TOC

+a k 11. 0 g 30. 0 o 6. 0 kg +a Find acceleration and tension Step 1 - Guess the direction of acceleration. Let’s guess this way. (it’s wrong) Hmmm. Coconuts? W

+a k 11. 0 g 30. 0 o 6. 0 kg +a Find acceleration and tension Step 2 - Set up the equation for the 6. 0 kg mass. T is positive (up), and the weight of the mass is down Weight = (6. 0 kg)(9. 81 N/kg) = 58. 86 N down (-) Tension T is up (+), so we have T - 58. 86 N = (6. 0 kg)a W

+a k 11. 0 g 30. 0 o 6. 0 kg +a Find acceleration and tension Step 3 - Set up the equation for the 11. 0 kg mass. Remember, down the plane is now positive. You have the tension T up (-) the plane, and the parallel component of gravity down (+) the plane: F|| = mgsin( ) = (11. 0 kg)(9. 81 N/kg)sin(30. 0 o) = 53. 9550 N down (+) the plane, Tension T is up the plane (-), so we have: 53. 9550 N - T = (11. 0 kg)a W

+a k 11. 0 g 30. 0 o 6. 0 kg +a Find acceleration and tension Step 4 - Solve the math for the acceleration: 53. 9550 N - T = (11. 0 kg)a T - 58. 86 N = (6. 0 kg)a 53. 9550 N - T = (11. 0 kg)a +T - 58. 86 N = (6. 0 kg)a 53. 9550 N - T + T - 58. 86 N = (17. 0 kg)a 53. 9550 N - 58. 86 N = (17. 0 kg)a a = -0. 288529412 m/s/s We guessed wrong!! it accelerates the other way!!! a = -0. 29 m/s/s W

+a k 11. 0 g 30. 0 o 6. 0 kg +a Find acceleration and tension Step 5 - Solve for the tension: 53. 9550 N - T = (11. 0 kg)a T - 58. 86 N = (6. 0 kg)a a = -0. 288529412 m/s/s Plug into one of the equations: T - 58. 86 N = (6. 0 kg)a T = 58. 86 N + (6. 0 kg)(-0. 288529412 m/s/s) = 57. 12882 N 57 N W