Posted: August 4th, 2015

Physics

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!