2. Pulleys Example Whiteboards - Atwood’s Machine Whiteboards - Inclined Plane M1M1 M2M2

3. Example - Find Tension, and acceleration if  k =.11: 32.0 kg 5.0 kg Step 2 - Calculate or express = ma for every mass object For the 5.0 kg mass, the tension T is up (-) T (-) +49 N The weight of 5*9.8 = 49 N is down (+): And our formula becomes: 49 N - T = (5.0 kg)a Step 1 - guess direction of + acceleration (This becomes the + direction for each mass) +a

4. Example - Find Tension, and acceleration if  k =.11: 32.0 kg 5.0 kg Step 1 - guess direction of + acceleration (This becomes the + direction for each mass) +a Step 2 - Calculate or express = ma for every mass object 49 N - T = (5.0 kg)a The 32.0 kg mass has the tension T to the right (+) T And, assuming it is moving to the right, a frictional force of.11*32*9.8 = 34.496 N to the left (-) -34.496 N So our equation becomes: T - 34.496 N = (32.0 kg)a

5. And now it’s math time!!!! Two equations, two unknowns! 49 N - T = (5.0 kg)a T - 34.496 N = (32.0 kg)a 49 N - T + T - 34.496 N = (5.0 kg)a + (32.0 kg)a 49 N - 34.496 N = (37 kg)a a = (49 N - 34.496 N)/37 kg =.392 m/s/s And plug into an equation to find T: T - 34.496 N = (32.0 kg)a T = (32.0 kg)(.392 m/s/s) + 34.496 N = 47.04 N + Add them Together!

6. Whiteboards: Atwood’s Machine 11 | 2 | 3 | 4 | 52345 TOC

7. Find acceleration and tension Green, Bananas never do W 5.0 kg 3.0 kg Massless frictionless pulley Step 1 - Guess the direction of acceleration - This becomes the positive direction for each mass. Uhh well um. the 5.0 kg is heavier. +a

8. Find acceleration and tension T - 29.4 N = (3.0 kg)a W 5.0 kg 3.0 kg Massless frictionless pulley Step 2 - Set up the =ma for the 3.0 kg mass: T is up, and the weight is down. Down is - and up is + weight = (3.0 kg)(9.8 N/kg) = 29.4 N down (- in this case) T is up, -29.4 is down: T - 29.4 N = (3.0 kg)a +a

9. Find acceleration and tension 49 N - T = (5.0 kg)a W 5.0 kg 3.0 kg Massless frictionless pulley Step 3 - Set up the =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)(9.8 N/kg) = 49 N down (+ in this case) T is up (-), 49 N is down (+): 49 N - T = (5.0 kg)a +a

10. Find acceleration and tension 2.45 m/s/s W 5.0 kg 3.0 kg Massless frictionless pulley Step 4 - Solve for acceleration: 49 N - T = (5.0 kg)a T - 29.4 N = (3.0 kg)a 49 N - T = (5.0 kg)a +T - 29.4 N = (3.0 kg)a T - 29.4 N + 49 N - T = (8.0 kg)a a = 2.45 m/s/s +a

11. Find acceleration and tension 37 N W 5.0 kg 3.0 kg Massless frictionless pulley Step 5- Solve for T: 49 N - T = (5.0 kg)a T - 29.4 N = (3.0 kg)a a = 2.45 m/s/s Pick one of the formulas, and plug the acceleration in: T - 29.4 N = (3.0 kg)a T = (3.0 kg)(2.45 m/s/s) + 29.4 N = 36.75 N = 37 N +a

12. Whiteboards: Pulleys on Inclined Planes 11 | 2 | 3 | 4 | 52345 TOC

13. Find acceleration and tension Hmmm. Coconuts? W Step 1 - Guess the direction of acceleration. Let’s guess this way. (it’s wrong) 6.0 kg 11.0 kg  s = 0  k = 0 30.0 o +a

14. Find acceleration and tension T - 58.8 N = (6.0 kg)a Step 2 - Set up the equation for the 6.0 kg mass. T is positive (up), and the weight of the mass is down 6.0 kg 11.0 kg  s = 0  k = 0 30.0 o +a Weight = (6.0 kg)(9.8 N/kg) = 58.8 N down (-) Tension T is up (+), so we have T - 58.8 N = (6.0 kg)a W

15. Find acceleration and tension 53.9 N - T = (11.0 kg)a 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: 6.0 kg 11.0 kg  s = 0  k = 0 30.0 o +a F || = mgsin(  ) = (11.0 kg)(9.8 N/kg)sin(30.0 o ) = 53.9 N down (+) the plane, Tension T is up the plane (-), so we have: 53.9 N - T = (11.0 kg)a W

16. Find acceleration and tension a = -0.2882 m/s/s Step 4 - Solve the math for the acceleration: 53.9 N - T = (11.0 kg)a T - 58.8 N = (6.0 kg)a 6.0 kg 11.0 kg  s = 0  k = 0 30.0 o +a 53.9 N - T = (11.0 kg)a +T - 58.8 N = (6.0 kg)a 53.9 N - T + T - 58.8 N = (17.0 kg)a 53.9 N - 58.8 N = (17.0 kg)a a = -0.2882 m/s/s We guessed wrong!! it accelerates the other way!!! W

17. Find acceleration and tension 57.1 N Step 5 - Solve for the tension: 53.9 N - T = (11.0 kg)a T - 58.8 N = (6.0 kg)a a = -0.2882 m/s/s 6.0 kg 11.0 kg  s = 0  k = 0 30.0 o +a Plug into one of the equations: T - 58.8 N = (6.0 kg)a T = 58.8 N + (6.0 kg)(-0.2882 m/s/s) = 57.1 N W