Assignment:
Part 1
1. Liverpool played 38 games and obtained 84 points in a particular season. You obtain no points for losing a game, one point for drawing a game and three points if you win a game. If Liverpool won twice as many games as they lost, how many games did they win, draw and lose?
2. Consider a series of integers that are all positive and all of the numbers are taken to be less than 125. There are 43 differences between adjacent numbers in this series defined as, . Can you prove that some value of the differences (which must also be positive integers) must occur at least 10 times?
3. Jane is walking her dog, Spot. She sees her friend, Dick, walking toward her along the same long, straight road. Both Dick and Jane are walking at 3 mph. When Dick and Jane are 600 feet apart, Spot runs from Dick to Jane, turns and runs back to Dick, and then back and forth between them at a constant speed of 8 mph. Dick and Jane both continue walking toward each other at a constant 3 mph. Neglecting the time lost each time Spot reverses direction, how far has Spot run in the time it takes Dick and Jane to meet?
Part 2
1. Here are the densities and radii of planets in our solar system (in grams per cm3 and km, respectively)
Mercury 5.4 g cm-3 2,440 km
Venus 5.2 g cm- 6,052 km
Earth 5.5 g cm-3 6,378 km
Mars 3.9 g cm-3 3,396 km
Jupiter 1.3 g cm-3 71,492 km
Saturn 0.7 g cm-3 60,268 km
Uranus 1.3 g cm-3 25,559 km
Neptune 1.6 g cm-3 24,764 km
How much would you weigh on these different planets? 1 pound is equal to 0.453592 kg.
2. Consider an Atwood's machine consisting of two masses connected by a string which is over a pulley, as shown in the picture.
a) Draw a free body diagram for both block A and block B.
b) Apply Newton's second law to both block A and block B.
c) the mass of block A is 14 kg while the mass of block B is 6 kg, find the tension in the rope.
3. Consider a modified Atwood machine, with a mass sitting on a table connected by a string which stretches over a pulley connected to a hanging mass. Friction is ignored.
a) Draw a free body diagram for both block 1 and block 2.
b) Apply Newton's second law to both block 1 and block 2.
c) If the mass of block 2 is 20 kg while the mass of block 1 is 5 kg, find the tension in the rope.
Part 3
1. Consider a modified Atwood machine, with a mass sitting on an inclined plane connected by a string which stretches over a pulley connected to a hanging mass. See picture. Friction is ignored.
a) Draw a free body diagram for both block 1 and block 2.
b) Apply Newton's second law to both block 1 and block 2.
c) If the angle is 30o , and the mass of block 2 is 10 kg while the mass of block 1 is 15 kg, find the tension in the rope.
2. Consider the following system with a 2 kg mass sitting on a table, connected to both 1 kg and 3 kg masses by strings which go over two separate pulleys and are pulling on the 2 kg mass in opposite directions.
a) What static friction coefficient would be required between the 2 kg mass and the table to stop the masses from accelerating?
b) If the masses are in motion, what kinetic friction coefficient would be required between the 2 kg mass and the table to ensure the masses move at a constant velocity?
c) Find the tension in both pieces of string for the case of zero friction.
3. Tarzan is running towards a cliff, ready to swing down rescue Jane from some poisonous snakes, and make it back up to a tree on the other side. For now, we are going to assume energy is conserved and ignore momentum. (We'll come back to this problem later!)
If the height of the cliff is 10 m, the height of the tree on which Tarzan hopes to land is 7 m, the mass of Tarzan is 100 kg and the mass of Jane is 50 kg. What velocity does Tarzan have to be running to rescue Jane?
