Scaling. Vector Addition 5.)

Transcription

1 Scaling A statue is to be scaled down, without distorting its shape, by changing its total volume from 1.25 m 3 to 0.37 m 3. Explain your reasoning in each of the following calculations. A. If the height of the original statue is 250 cm, calculate the height of the smaller model. B. If the circular base of the original statue has a circumference of 45 cm, calculate the circumference of the scaled-down base in the smaller model. C. How will the total surface area of the model compare (this means an appropriate ratio) with the total surface area of the original? How will the surface areas of the circular bases compare? D. If both the model and the original are made of the same material, how will the mass of the model compare (again, this means your answer should describe the appropriate ratio) with the mass of the original? E. If the model and the original are not made of the same material, what would you have to know about the materials to be able to compare the masses, and how would you use this information? F. If the original statue and the model turned out to have the same mass, what would you conclude about the materials making up the two objects? (Give a numerical answer comparing relevant properties of the materials.) Vector Addition 5.) cm@ ii) A pet-store supply truck moves at 25m/s north along a highway. Inside, a dog moves at 5.75m/s at an angle of 35 degrees East of North. What is the velocity of the dog relative to the road? 12. ii) Humanoid robots designed to replace us are being produced in a faraway factory. The robots are moved along a conveyor belt which moves at 3m/s at 65 relative to a wall. One robot decides to walk on the conveyor at 45, at 5m/s. What is its resultant velocity? 13. iii) A plane is flying from Boston to New York, 200 miles away, departs at 1:00 and is scheduled to arrive at 1:45. (Pretend NYC is directly south of Boston.) A crosswind blowing to the west with a speed of 20 miles/hour would push the plane off course. The pilot, who had of course taken physics, knows that if he aims at an angle towards the wind, he will still make it to New York. At what velocity and angle should the pilot direct the plane? 14. iii)the current of a 200m wide straight river has a flow rate of 2.5km/h. A motorboat with a speed of 30km/h in still water crosses the river. Aim the boat 45 degrees up river. Where does it land? Static Equilibrium Determine the weight of the ball if the system is in equilibrium. The cable at right exerts a 30. N force.

2 In the system below the pulleys are frictionless and the system hangs at static equilibrium. If w1, the weight of the object on the right, is 200. N, what are the values of w2 and w3? Torque Problem 8, pg Problem 21, pg Big Fred is an artist, and has an idea to make a balancing piece of artwork. He s got a 5m long metal plank, m=100kg, and on the far right end, he will place a 50kg stone statue. Where should the fulcrum be so this thing will balance? COM The earth has a mass of 6 x kg. The moon s mass is 7.36 x kg. The center-center distance between them is about 382,500km (on average). How far from the earth s center is the CoM of the earth-moon system? Is this point inside the earth? a) A meter long rod of uniform density has a mass of 2kg. Where is the center of mass of this rod? b) If an apple with a mass of 1kg, is attached to one end of the meter long rod, where is the new center of mass? There exists a square metal sheet of uniform density and a mass of 10kg. On three corners of this sheet, each 4cm from the center, a mass is placed. The masses in clockwise order from the bottom left are 3kg, 4kg, and 3 kg. Where is the CoM? Find the center of mass of this figure: Materials Science Ch , 45, 49, 55, 58 Kinematics A German stuntman named Martin Blume performed a stunt called the wall of death. To perform it, Blume rode his motorcycle for seven straight hours on the wall of a vertical cylinder Suppose that in a time interval of 30s Blume increases his speed steadily from 30 km hr to 45 km hr while circling inside the cylindrical wall. If the cylinder s diameter is 10m, how many laps does Blume make during the 30s time interval? Peter Rosendahl rode his unicycle a distance of 100m in 12.11s. If he started at rest, what was the magnitude of his acceleration?

3 In 1976, Kitty Hambleton of the United States drove a rocket engine car to a maximum speed of 965 km hr. Suppose Hambleton started at rest and underwent a constant acceleration of 4 m s 2. What distance would she have had to travel in order to reach the maximum speed? With a cruising speed of 2300 km hr, the French supersonic passenger jet Concorde is the fastest commercial airplane. Suppose the landing speed is 20% of the cruising speed. If the brakes can produce a deceleration of 5.8 m s 2, what minimum length of runway is necessary? 18. A helicopter is ascending vertically with a speed of 8 m s. At a height of 120m above the ground, a package is dropped from a window. How much time does it take for the package to reach the ground? 20. Pelicans tuck their wings and free fall straight down when diving for fish. Suppose a pelican starts its dive from a height of 20m and cannot change its path once committed. If it takes a fish 0.10s to perform evasive action, at what minimum height must it spot the pelican to escape? Assume the fish is at the surface of the water. Forces, N2L A box of moldy raisins is tossed angrily across a desktop, and goes from 4m/s to rest as it slides 1.5 meters. Its mass is 25 grams. What is the friction force between it and the table? Grandpa hasn t driven a car in a while, and he s a bit senile, so he uses the gas pedal and the brakes at the same time, thinking they re both gas pedals. So he guns it, and the engine provides 5400N of forward force, while the brakes offer 4400N of friction. For a 2000kg car, how long will it take to get up to highway speed, 27 s m? Friction Part I Horizontal. For each situation, determine if the block moves. If the object does move, give the kinetic friction force and find the acceleration (with direction). Tip: First find N, f s max and f k. If Applied force > f s max, it will move. Find the resulting acceleration. 1. Mass=20kg, µ s =.3, µ k = Mass=10kg, µ s =.22, µ k =.14 50N 60 30N 20 N 10N Part II. Vertical. For each situation, determine if the block moves. If not, give the actual friction force acting. If the object does move, give the actual friction force and find the acceleration (with direction). Tip: Steps to follow: First find N, f s max and f k. If all-the-forces-parallel-to-the-surface > f s max, it will move. Find acceleration

4 4. Mass=10kg, µ s =.22, µ k = N 5. P 104, #39 6. P 104, #40 Your mass is 55kg. Riding in an elevator, accelerating upward, the bathroom scale you are standing on reads 700N. What is your acceleration? Once you are moving at constant speed, what will the scale read? As you near the top, the scale s reading drops to 400N. What is the magnitude and direction of your acceleration? You are sitting at the top of a 3m long slide angled at 30 to the ground. If the µ is.11, how fast will you be going at the end of the slide? Extra challenge: can you solve this using work and energy? (it s actually much easier!) MOI, Rotational motion Mammals that depend on being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body. On the basis of rotational dynamics, explain why this distribution of mass is advantageous. Explain the physics justification for choking up on a baseball bat. If there were a great migration of people toward the equator, how would this affect the length of a day? There are 2 meter sticks with clamps on them. One has the masses at the ends, the other has the masses near the middle. Hold one in each hand and try to twist them back and forth. What do you observe? Explain. 9. On the record player, rotating at 33 rpm, a paperclip sits at the edge of the record (r=15cm). a. What is the angular velocity of the paperclip (rad/s)? b. What is the linear velocity of the paperclip? c. A second paperclip is 7.5cm from the center. What is its angular velocity? d. What is its linear velocity? e. If it takes 1/3 revolution for the record to reach the proper speed from rest, what is its angular acceleration? Circular motion Ch 5, probs 2, 7, 9, 12, 15

5 Energy, Work and Power 1. A spring loaded pea-shooter has a spring with k = 25N/m is compressed 10cm. A 5 gram pea is fired. How fast will it be going when it leaves the barrel? 2. Ricardo the claymation boy (mass = 5kg) is dropped from an airplane 5000 m above the earth. He pulls his chute and lands gracefully on a bed of downy softness with a speed of 5m/s. How much GPE did he have initially? How much KE did he land with? How much energy was dissipated as heat during the descent? 3. A) A roller coaster takes riders up 60 meters above the ground, and releases them. The total mass of the cars and riders is 400kg. Find the speed at points A, B, and C of the coaster. (Assume no friction) 60m A 15m B 30m C B) Suppose the track just ends at point C, and the cars go skidding along the ground. If μ between the cars and the ground is.3, how far will the cars go before stopping? C) Suppose a very large spring was used to launch the cart up to the top of the ramp, where it just barely makes it over the hill (velocity reaches near zero at the top). If the spring has a constant k of 54000N/m, how far must the spring be compressed to get the cart to the top of the hill. 7. A typical AA battery is rated at 1200mW-hr of energy. If you can shoot 400 pictures on one set of batteries, how much energy is the camera using per shot? The net force (in newtons) acting on a puck on an air table (an essentially frictionless system) varies with distance as shown in the following diagram: Magnitude of F (N) Position X (m) The puck has a mass of kg. When the force is applied to the puck at clock reading t = 0.00 the puck has an initial instantaneous velocity of m/s in the positive direction. A. Describe, in terms of work and kinetic energy, what happens to the puck over the space interval between x = 0m and x = 0.6m. Be specific. Do not use Newton s Second Law. The concepts and relationships between quantities that you describe here will guide you calculations in the following parts. B. Calculate the work that must have been done by the net force on the puck. C. Calculate the change in kinetic energy of the puck. D. Calculate the initial and final values of the kinetic energy of the puck E. How fast is the puck moving at 0.6m?

6 Momentum A 13 g bullet traveling at 330m/s penetrates a 2.0 kg block of wood and emerges going 270 m/s. If the block is stationary on a frictionless surface when hit, how fast does it move after the bullet emerges? A 115 kg fullback is running at 4.0 m/s to the east and is stopped in.75s by a head-on tackle by a tackler running due west. Calculate (a) the original momentum of the fullback, (b) the impulse exerted on the fullback, (d) the impulse exerted on the tackler, and (d) the average force exerted on the tackler. Suppose the force acting on a tennis ball (mass kg) as a function of time is given by this graph. Use graphical methods to estimate (a) the total impulse given the ball, and (b) the speed of the ball after being struck. Assuming the ball is being served so it is nearly at rest initially. Use figure 7-27 on p.188 Fluids, static pressure 1. An office window has dimensions 3.0 m by 1.8 m. As a result of the passage of a storm, the outside air pressure drops to 0.97 atm, but inside the pressure is held at 1.0 atm. What net force pushes out on the window? 2. Calculate the hydrostatic difference in blood pressure between the brain and the foot in a person of height 1.73m. The density of blood is 1.06 x 10 3 kg/m 3. Fluids in motion, Bernoulli 1. Your boat weighs 1200N. What volume of water will it displace when floating motionless at the surface of a lake? What is the mass of this water? 2. Your town is installing a fountain in the main square. If the water is to rise 25m above the fountain, how much pressure must the water have as it moves slowly toward the nozzle that sprays it up into the air? 3. Rather than putting a pump in the fountain, the town engineer puts a water storage tank in one of the nearby high rise office buildings. How high up in that building should the tank be for its water to rise to 25m when spraying out of the fountain? 4. Ch 10, #29 (p 291) Poiseuille s Law 1. Ch 10 # 37 (p292) 2. #40 3. #41 4. #44 5. When your friend s house was new, the kitchen faucet could deliver.5 liters per second. But mineral deposits have built up in the pipes over the years and reduced their effective diameters by 20%. How much water can the faucet deliver now? 6. How much higher must your blood pressure get to compensate for a 5% narrowing in your blood vessels? 7. How quickly would you have to move a 1 cm diameter stick through olive oil to reach a Reynolds number of 2000, so that you would begin to see turbulence around the stick? (Olive oil has a density of 918kg/m 3 ) 8. The effective obstacle length of a blimp is its width the distance to which the air is separated as it flows around the blimp. How slowly would a 15m wide blimp have to move in order to keep the airflow around it laminar? (density of air = 1.25kg/m 3 )

7 Projectiles 5. A hunter lying on the ground fires his gun at an angle of 2.2 at a final exam in a tree. The tree is 1500m away, and the test is 57.6m up in the tree. The bullet leaves the gun at 1250 m s. a) If the test lets go at the same moment of the gunfire, how far off the ground will it be when the bullet gets that far? b) How high up will the bullet be? Will the test be ok? (brainteaser: what subject is the test on?) 6. Sam S. hits a baseball 110 meters from home plate, which hits Mr. Anderson s front bumper, leaving a little paint there. a) If the ball was in the air for 3.5 seconds, with what speed did the ball leave his bat? b) How high did it go? (Assume the original and final heights are the same). 7. A football is kicked at ground level with a speed of 20 m/s at an angle of 37 degrees to the horizontal. How much later does it hit the ground? 8. Mr. A launches off a big table top (a type of jump) while snowboarding. The jump is angled at 30. The lip of the jump is 1 meter above the landing level. The distance cleared is 8 meters. How fast was he going? 30 1m 8m