3.Q1) (15 points) In the diagram below, M1 = 50 Kg, M2 = 20 Kg, mass and radius of the
g and 30 cm each, respectively. Both M1 and M2 rest on frictionless surfaces and the system starts from rest.
(a) Draw the fbd for each of M1, M2 and the pulley.
(b) Write the equations of motion for each of M1, M2 and the pulley.
(c) Calculate the linear acceleration of the two masses, as well as the angular acceleration of the pulley.
(d) Calculate the angular velocity of the pulley after M1 and M2 have been displaced linearly by 2 m.
Q2) (10 points) A basketball is thrown with an initial speed v0 of 10.8 m/s at 400 above horizontal, and it enters the hoop from above. The ball is released at 2.00 m above the ground. The hoop is 3.05 m above the ground and 10.0 m away from the player.
(a) Find the time at which the ball passes through the hoop.
(b) Find the ball’s velocity (express in component form) just when it enters the hoop.
(c) Find the ball’s maximum height.
Q3) (5 Points) An object is thrown up from the top of a building of height of 400 m with an initial velocity of 20 m/s.
(a) Find the position and the velocity of the object 5 s later.
(b) With what velocity will it hit the ground?
(c) At the same time the first object is thrown up, a second object is thrown up from the ground at 100 m/s. Will the two objects collide? If yes, calculate when and where,
6.You make very good pizzas, so you decide to sell your pizzas on campus. Since the set up for making
pizza is already available to you, the only cost involved is that of making the pizza, which you calculate to be $ 5 per pizza.
a. What is the cost function?
If 10 pizzas are available in a day, the market offers a price of $ 11 per pizza. If 50 pizzas are available in a day, the market offers a price of $ 7 per pizza.
b. Assuming a linear relationship between price and quantity, find the price that the market offers as a function of the number of pizzas available. You start selling the pizzas.
c. What is revenue as a function of the quantity you sell? What is the profit function?
d. What quantity will maximize your profit? Call it q ∗ 1. What is the maximum profit?
e. If somebody is already supplying 5 pizzas every day, What is the maximum profit that you can make?
9.Suppose that the economy is at full employment (our economy has reached its potential GDP or the maximum that we
imum that we can normally produce). Now suppose that the Federal Government decides to decrease taxes. If we compare the long run price and GDP levels to the price and GDP levels that existed before the Federal Government’s action, we would find that_______?
Production or the GDP would not increase in the long run.
Prices would decrease in the long run.
A decrease in unemployment would result in the long run.
Producers would increase production in the long run as a result of the Federal Government’s actions
10.A uniform beam of length L
and mass m shown in Figure
P12.16 is inclined at an angle
u to the horizontal. Its
izontal. Its upper
end is connected to a wall by
a rope, and its lower end rests
on a rough, horizontal sur-
face. The coefficient of static
friction between the beam
and surface is ms. Assume
the angle u is such that the static friction force is at its
maximum value. (a) Draw a force diagram for the beam.
(b) Using the condition of rotational equilibrium,
find an expression for the tension T in the rope in
terms of m, g, and u. (c) Using the condition of trans-
lational equilibrium, find a second expression for T in
terms of ms, m, and g. (d) Using the results from parts
(a) through (c), obtain an expression for ms
vertical component of this force. Now solve the same
problem from the force diagram from part (a) by com-
puting torques around the junction between the cable
and the beam at the right-hand end of the beam. Find
(e) the vertical component of the force exerted by the
pole on the beam, (f) the tension in the cable, and
(g) the horizontal component of the force exerted
by the pole on the beam. (h) Compare the solution
to parts (b) through (d) with the solution to parts
(e) through (g). Is either solution more accurate?
19. Sir Lost-a-Lot dons his armor and sets out from the
castle on his trusty steed (Fig. P12.19). Usually, the
drawbridge is lowered to a horizontal position so that
the end of the bridge rests on the stone ledge. Unfor-
squire didn’t lower the draw-
ing only the angle u. (e) What happens if the ladder
is lifted upward and its base is placed back on the
ground slightly to the left of its position in Figure