1. Using the “Clicker” If you have a clicker now, and did not do this last time, please enter your ID in your clicker. First, turn on your clicker by sliding the power switch, on the left, up. Next, store your student number in the clicker. You only have to do this once. Press the * button to enter the setup menu. Press the up arrow button to get to ID Press the big green arrow key Press the T button, then the up arrow to get a U Enter the rest of your BU ID. Press the big green arrow key.

2. Baseball and basketball The two balls drop through the same height h, so they are both traveling down with a speed v when the basketball hits the ground. Assumption: the basketball’s collision with the floor is elastic, so after colliding with the floor the basketball’s velocity is v directed up.

3. Baseball and basketball Now we have a collision between the baseball, of mass m and velocity v down, and the basketball, of mass 3m and velocity v up. Observation: What does the basketball do after the collision?

4. Baseball and basketball Now we have a collision between the baseball, of mass m and velocity v down, and the basketball, of mass 3m and velocity v up. Observation: What does the basketball do after the collision? The basketball is essentially at rest after the collision. Apply momentum conservation to find the speed of the baseball. Choose up to be positive. v f = ?

5. Baseball and basketball Now we have a collision between the baseball, of mass m and velocity v down, and the basketball, of mass 3m and velocity v up. Observation: What does the basketball do after the collision? The basketball is essentially at rest after the collision. Apply momentum conservation to find the speed of the baseball. Choose up to be positive. v f = +2v (this solution also conserves kinetic energy)

6. Baseball and basketball How high does the baseball go? When it was dropped from a height h, the baseball acquired a speed v: If the baseball is fired up with a speed 2v, how high does it go?

7. Baseball and basketball How high does the baseball go? When it was dropped from a height h, the baseball acquired a speed v: If the baseball is fired up with a speed 2v, how high does it go? H = 4h. The baseball goes four times as high. It would go even higher, to 9h, if it’s mass was negligible compared to the basketball’s mass.

8. Collisions in two dimensions The Law of Conservation of Momentum applies in two and three dimensions, too. To apply it in 2-D, split the momentum into x and y components and keep them separate. Write out two conservation of momentum equations, one for the x direction and one for the y direction. For example: Simulation of a 2-D collision

9. Fluids We now turn to exploring fluids. Some of this will be new, but much of it will involve applications of familiar ideas like Newton’s second law, or energy conservation. What is a fluid?

10. Fluids We now turn to exploring fluids. Some of this will be new, but much of it will involve applications of familiar ideas like Newton’s second law, or energy conservation. What is a fluid? A fluid is something that can flow. A fluid can be a liquid or a gas.

11. The Buoyant Force With fluids, we bring in a new force. The buoyant force is generally an upward force exerted by a fluid on an object that is either fully or partly immersed in that fluid. Let’s survey your initial ideas about the buoyant force.

12. The Buoyant Force A wooden block with a weight of 100 N floats exactly 50% submerged in a particular fluid. The upward buoyant force exerted on the block by the fluid … 1. has a magnitude of 100 N 2. has a magnitude of 50 N 3. depends on the density of the fluid 4. depends on the density of the block 5. depends on both the density of the fluid and the density of the block

13. Learning by Analogy Our 100 N block is at rest on a flat table. What is the normal force exerted on the block by the table?

14. Learning by Analogy Our 100 N block is at rest on a flat table. What is the normal force exerted on the block by the table? To answer this, we apply Newton’s Second Law. There is no acceleration, so the forces balance.

15. Learning by Analogy Apply the same method when the block floats in the fluid. What is the magnitude of the buoyant force acting on the block?

16. Learning by Analogy Apply the same method when the block floats in the fluid. What is the magnitude of the buoyant force acting on the block? To answer this, we apply Newton’s Second Law. There is no acceleration, so the forces balance.

17. Reviewing the normal force We stack a 50-newton weight on top of the 100 N block. What is the normal force exerted on the block by the table?

18. Reviewing the normal force We stack a 50-newton weight on top of the 100 N block. What is the normal force exerted on the block by the table? To answer this, we apply Newton’s Second Law. There is no acceleration, so the forces balance. The block presses down farther into the table (this is hard to see).

19. Applying this to the buoyant force We stack a 50-newton weight on top of the 100 N block. What is the buoyant force exerted on the block by the fluid?

20. Applying this to the buoyant force We stack a 50-newton weight on top of the 100 N block. What is the buoyant force exerted on the block by the fluid? To answer this, we apply Newton’s Second Law. There is no acceleration, so the forces balance. The block presses down farther into the fluid (this is easy to see).

21. Apply Newton’s Second Law Even though we are dealing with a new topic, fluids, we can still apply Newton’s second law to find the buoyant force.

22. Three Beakers The wooden block, with a weight of 100 N, floats in all three of the following cases, but a different percentage of the block is submerged in each case. In which case does the block experience the largest buoyant force? 4. The buoyant force is equal in all three cases.

23. Three Beakers What does the free-body diagram of the block look like? What is the difference between these fluids?

24. Three Beakers What does the free-body diagram of the block look like? What is the difference between these fluids? The density.

25. Worksheet, page 1 A block of weight mg = 45.0 N has part of its volume submerged in a beaker of water. The block is partially supported by a string of fixed length. When 80.0% of the block’s volume is submerged, the tension in the string is 5.00 N. What is the magnitude of the buoyant force acting on the block? Hint: try drawing a free-body diagram of the block and applying Newton’s Second Law.

26. Apply Newton’s Second Law The block is in equilibrium – all the forces balance. Taking up to be positive:

27. Worksheet, page 1 Water is steadily removed from the beaker, causing the block to become less submerged. The string breaks when its tension exceeds 35.0 N. What percent of the block’s volume is submerged at the moment the string breaks? Hint: try drawing a free-body diagram of the block and applying Newton’s Second Law to first find the buoyant force.

28. Apply Newton’s Second Law The block is in equilibrium – all the forces balance. Taking up to be positive: The buoyant force is proportional to the volume of fluid displaced by the block. If the buoyant force is 40 N when 80% of the block is submerged, when the buoyant force is 10 N we must have __% of the block submerged.

29. Apply Newton’s Second Law The block is in equilibrium – all the forces balance. Taking up to be positive: The buoyant force is proportional to the volume of fluid displaced by the block. If the buoyant force is 40 N when 80% of the block is submerged, when the buoyant force is 10 N we must have 20% of the block submerged.

30. Worksheet, page 1 After the string breaks and the block comes to a new equilibrium position in the beaker, what percent of the block’s volume is submerged? Hint: what does the free-body diagram look like now?

31. Apply Newton’s Second Law The block is in equilibrium – all the forces balance. Taking up to be positive: The buoyant force is proportional to the volume of fluid displaced by the block. If the buoyant force is 40 N when 80% of the block is submerged, when the buoyant force is 45 N we must have __% of the block submerged.

32. Apply Newton’s Second Law The block is in equilibrium – all the forces balance. Taking up to be positive: The buoyant force is proportional to the volume of fluid displaced by the block. If the buoyant force is 40 N when 80% of the block is submerged, when the buoyant force is 45 N we must have 90% of the block submerged.

33. What does the Buoyant Force depend on? Let’s do some experiments to see what the buoyant force depends on. We’ll prepare a beaker, filled with fluid up to an overflow spout, and then place a block in the beaker. The overflow spout drains into another beaker sitting on a scale – the scale is tared to read zero.

34. Experiment 1 Place a particular block with a weight of 8 N in the beaker. It floats, 40% submerged. We observe that the block displaces a volume of fluid that has a weight of 8 N.

35. Experiment 2 Place a block of the same size, but with a weight of 16 N, in the beaker. It floats, 80% submerged. We observe that the block displaces a volume of fluid that has a weight of 16 N. The buoyant force has doubled, and the volume of fluid displaced by the block has doubled.

36. Hypothesis The buoyant force acting on an object is proportional to the volume of fluid displaced by that object.

37. Experiment 3 Place a block of the same size, but with a weight of 24 N, in the beaker. It sinks, so we suspend it from a spring scale. What is the buoyant force, in this case?

38. Experiment 3 Place a block of the same size, but with a weight of 24 N, in the beaker. It sinks, so we suspend it from a spring scale. The buoyant force is 20 N up, to balance the forces. 25% more fluid displaced than experiment 2, 25% larger buoyant force.

39. Archimedes’ Principle This is true - the buoyant force acting on an object is proportional to the volume of fluid displaced by that object. But, we can say more than that. The buoyant force acting on an object is equal to the weight of fluid displaced by that object. This is Archimedes’ Principle.

40. A Floating Object When an object floats in a fluid, the downward force of gravity acting on the object is balanced by the upward buoyant force. Looking at the fraction of the object submerged in the fluid tells us how the density of the object compares to that of the fluid.

41. Beaker on a Balance A beaker of water sits on a scale. If you dip your little finger into the water, what happens to the scale reading? Assume that no water spills from the beaker in this process. 1. The scale reading goes up 2. The scale reading goes down 3. The scale reading stays the same

42. Three Blocks We have three cubes of identical volume but different density. We also have a container of fluid. The density of Cube A is less than the density of the fluid; the density of Cube B is exactly equal to the density of the fluid; and the density of Cube C is greater than the density of the fluid. When these objects are all completely submerged in the fluid, as shown, which cube displaces the largest volume of fluid? 1. Cube A 2. Cube B 3. Cube C 4. The cubes all displace equal volumes of fluid

43. Each cube displaces a volume of fluid equal to its own volume, and the cube volumes are equal so the volumes of fluid displaced are all equal. Three Blocks

44. Which object has the largest buoyant force acting on it? 1. Cube A 2. Cube B 3. Cube C 4. The cubes have equal buoyant forces

45. Each cube displaces an equal volume of the same fluid, so the buoyant force is the same on each. Three Blocks

46. Whiteboard