Posted: September 18th, 2017

Homework #1

1. Liverpool played 38 games and obtained 74 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?

Homework #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-3 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.

Homework #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?

Homework #4

1. Now you’ve covered momentum lets look again at last weeks

question. 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. Now we must remember that energy isn’t

conserved in the collision (of course it is, but some of the energy is

lost as heat, sound, etc… so its not recoverable), but momentum is!

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 at to rescue Jane?

2. You are in a small boat in a swimming pool and have a brick also in the boat with you. You throw the

brick overboard and into the swimming pool. Does the level of the water in the swimming pool rise,

stay the same or decrease? Show calculations to back up your answer!

3. In the olden days we used to have milk in bottles delivered by the milkman.

The cream would rise to the top of the bottom, and occasionally a bird would

come along and peck at the top of the bottle to get at the cream! Compare the

milk bottle where the cream is still uniformly mixed with the milk, to the same

milk bottle later on when the cream has separated and risen to the top. Does the

hydrostatic pressure at the bottom of the milk bottle remain the same during the

phase separation of the cream?

Homework #5

1. Its raining and your trying to decide whether or not to run or walk. If you walk you’ll spend longer in

the rain, but if you run you’ll be running into the rain and potentially get wetter. The rain is falling with

a velocity of which has both an x-component as it blows sideways and a y-component as it falls.

You move with a velocity of which only has an x-direction. If the rain is falling with a density

and you have a top area of (which rain will fall onto) and a sideways area of (which rain

will fall onto if it blows sideways or you move sideways into it), find an expression for how wet you

would be when traveling a distance d, as a function of your velocity.

2. The range equation (see Cycle 2: Lecture 3: Page 4) can only be used if the projectile is launched

from the ground and returns to the same height as it was fired from. What if instead it was fired from a

given height, h?

a) Find the range of an object fired from a height h and at an angle .

b) Find the angle of launch which maximizes the range as a function of height.

3. A skateboarder in a death-defying stunt decides to launch herself

from a ramp on a hill. The skateboarder leaves the ramp at a height

of 1.4 m above the slope, traveling 15 m s-1 and at an angle of 40o

to the horizontal. The slope is inclined at 45o to the horizontal.

a) How far down the slope does the skateboarder land?

b) How long is the skateboarder in the air?

c) With what velocity does the skateboarder land on the slope?

Homework #6

1. A car is driving around a bank curve such that it

can safely go around a circular curve at a given

velocity, vo, even when on ice (zero friction). Any

slower and the car would slide down towards the

center of the circle, any faster and the car would

slide up the hill and away from the center of the

circle. If a static frictional coefficient, , is

introduced then the car can safely navigate around

the curve at any speed between a minimum speed of

vmin and a maximum speed of vmax. Can you find

expressions for vmin and vmax as a function of vo,

and the radius of the road, R.

2. In the system shown three

masses are connected via string

and a series of pulleys to one

another.

Find the acceleration of all blocks.

Do not assume the slope to be

frictionless, but assume that block

A is in motion.

3. In a cardiac stress test the patient is required to walk on an inclined treadmill. Imagine that the

patients mass is 80 kg and that the inclined treadmill is at a slope of 15o. The efficiency of the human

body can be taken to be 25%.

a) Obtain an expression for the power required by the patient to maintain a velocity of 3 m s-1.

b) How long would the patient have to walk on the treadmill to burn the energy contained in a bottle of

beer, a slice of pizza and a jelly doughnut?

Homework #7

1. A 1 kg block of wood is moving with a velocity of 10 m s-1 on top of a table. The coefficient of

kinetic friction between the block of wood and the table is 0.1. The block of wood is a cube with sides

of 12 cm.

a) How far does the block move before coming to a stop?

b) If momentum id conserved, where did the momentum of the block go?

c) The block slows down because of the frictional force between the block and the block. However, if

the frictional force acts at the bottom of the block and the blocks center of mass is in the center of the

block then there must be a torque acting on the block. Calculate this torque. Why doesn’t this torque

cause the block to rotate?

2. In 2286, Admiral Kirk and his crew were forced to use the sling

shot effect in a stolen Klingon Bird-of-Prey to travel back in time

to the late 20th century to retrieve two humpback whales. The

stolen Klingon Bird-of-Prey traveled towards the sun at a velocity

of v, while the sun was moving towards them at a velocity of u,

then traveled around the sun (using the sun’s gravitational field)

such that the stolen Klingon Bird-of-Prey was now moving in the

opposite direction from whence it started, with a new velocity,

vnew. Find vnew in terms of u and v, assuming the mass of the sun is

much larger than the mass of the spaceship.

3. Consider holding a carton of milk in your hand, as shown in the image.

The force of the bicep muscle acts at an angle of 15o to the vertical, while

the weight of the arm and the milk both act downwards. The distance

from the elbow to where the bicep muscle is attached via the the distal

bicep tendons to the radius and ulna bones is 5 cm. The distance from the

elbow to the center of mass of the forearm is 16.5 cm and the distance

from the elbow to the hand, holding the milk, is 35 cm. The forearm has

a mass of 4 kg and the milk carton a mass of 2 kg.

a) Assuming the forearm is kept perfectly horizontal, find the tension in the bicep muscle.

b) As a function of the angle of the forearm with respect to the horizontal direction (as the forearm is

lowered) calculate the tension in the bicep muscle.

c) Include a plot of tension in the bicep as a function of the angle of the forearm relative to the

horizontal (don’t forget to label axis).

Homework #8

1. An aortic aneurysm exists as a bulging out of the aorta walls, where the aorta walls are actually more

elastic than normal aorta walls, and deform more in response to the blood pressure in the aorta. If the

radius of the aorta is typically 1 cm and the blood flow rate is 100 cm3 s-1, how much would the

pressure increase in the aortic aneurysm if the radius of the aneurysm is 3 cm? Assume that the blood

vessel is horizontal and ignore the viscous nature of blood.

2. You hold a hose at 45o to the horizontal and at a height of 1 m from the floor. The water reaches a

maximum distance of 10 m from where you are standing. Now you place your thumb over the end of

the hose to occlude the opening by 80%, which in turn reduces the flow rate by 50%. Even though less

fluid emerges the water travels further?

a) How far does the water travel with your thumb over the end of the hose (assuming the height and

angle remain the same)?

b) Assume the flow rate (as a percentage of original flow rate) can given by

where is the percentage by which the opening of the hose is occluded. Notice that if is 0% then

is 100%, whereas if is 100% then is 0%. Find the amount by which the hose must be

occluded in order for the water to travel twice as far (assuming the height and angle remain the same).

3. Assume the muscle is 37oC and is separated from the outside air by

layers of fat and skin. The layer of fat, at a particular location on the

skin, is 2 mm thick and has a conductivity of 0.16 W m-1 K-1. Above this

is a 1.1 mm dermis layer with a thermal conductivity of 0.53 W m-1 K-1.

Finally, the outermost epidermal layer is 0.1 mm thick with a thermal

conductivity of 0.21 W m-1 K-1. How much heat is lost per unit area and

unit time if the ambient air temperature is 0oC?

Homework #9

1. A ball is thrown from the top of a roof at an angle of 20o with respect to the vertical. 1 s later a ball is

dropped from the top of the roof.

a) If the height of the roof is 20 m, determine the velocity with which the first ball must be thrown in

order for both balls to land at exactly the same time.

b) Imagine the velocity of the first ball is now known, and instead it is the height of the building which

is unknown. Can you obtain an equation for the height of the building which would result in both balls

landing on the floor at the same time.

2. You are firing a cannonball from the top

of a cliff at a pirate ship which is behind a

mountain. The height of the cannon is 100

m, the height of the mountain is 120 m,

and the height of the pirate ship is 70 m.

The mountain is horizontally 200 m away

from cannon, while the pirate ship is

horizontally 270 m away from the cannon.

The speed of the cannonball after it is fired is 65 m s-1. You could hit anywhere from the bottom of the

pirate ship to the top in order to sink it. Over what angles could you fire the cannonball to sink the

pirate ship?

Homework #10

1. Consider the pulley system to the right. The masses M and

3 M are simply hanging, while the mass 2 M is sitting on an

inclined plane with coefficients of friction and .

a) Obtain an expression for the coefficient of static friction

required to ensure the masses do not accelerate, which

depends on the angle of the inclined surface.

b) Obtain an expression for the coefficient of kinetic friction

required to ensure the masses move at a constant velocity

when they do move, which depends on the angle of the

inclined surface.

2. A bullet of mass m and initial velocity v0 passes through a block of mass M suspended by an

unstretchable, massless string of length L from an overhead support. It emerges from the collision on

the far side traveling at v1 < v0. This happens extremely quickly (before the block has time to swing up)

and the mass of the block is unchanged by the passage of the bullet (the mass removed making the hole

is negligible, in other words). After the collision, the block swings up to a maximum angle θmax and then

stops. Find θmax.

3. In the figure above, a spool or pulley with moment of inertia

is hanging from a ceiling by a (massless, unstretchable) string

that is wrapped around it at a radius R, while a block of equal mass

M is hung on a second string that is wrapped around it at a radius r as

shown. Find the magnitude of the acceleration of the the central

pulley.

4. Write 5 multiple-choice exam questions on any topics of your choice. A good question should be of

sufficient difficulty and perhaps have a good background story? Try to change questions you’ve seen

previously? Post your favorite question to the blackboard discussion board!

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