Physics 1B – Midterm 1 solutions
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Name :
You have 1h30 to complete this exam. You are only allowed a standard calculator and an equation
sheet.
Don’t forget to put your ID and name on each page of your work (including extra paper). Please put
your exam in your section pile at the end of the exam. Thank you.
Exercise 1 (25 points)
University Physics Pearson 15th edition
12.73 A closed and elevated vertical cylindrical tank with diameter 2.00 m contains water to a depth
of 0.800 m. A worker accidently pokes a circular hole with diameter 0.0200 m in the bottom of the
tank. As the water drains from the tank, compressed air above the water in the tank maintains a gauge
pressure of 5.00 * 103 Pa at the surface of the water. Ignore any effects of viscosity.
(a) Draw the situation and explain the context (5 points)
(b) Just after the hole is made, what is the speed of the water as it emerges from the hole? (5 points).
What is the ratio of this speed to the efflux speed if the top of the tank is open to the air? (5 points)
(c) How much time does it take for all the water to drain from the tank? (5 points).
(d) What is the ratio of this time to the time it takes for the tank to drain if the top of the tank is open
to the air? (5 points)
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Exercise 2 (25 points)
University of physics 14th edition
15.18 A 1.5-m string of weight 0.0125 N is tied to the ceiling at its upper end, and the lower end
supports a weight W. Ignore the very small variation in tension along the length of the string that is
produced by the weight of the string. When you pluck the string slightly, the waves traveling up the
string obey the equation
830
(a) What is the amplitude, frequency and phase of the propagating wave? (5 points)
T
it
Ir
(b) How much time does it take a pulse to travel the full length of the string? (5) points)
(c) What is the weight W? Is it reasonable to assume that the tension in the string is constant?
Explain (5 points)
v
Fm
(d) How many wavelengths are on the string at any instant of time? (5 points)
1.5
2
V
0.551
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(e) What is the equation for waves traveling down the string?
Note: We assume there is no standing waves produced yet (5 points)
Exercise 3 (25 points)
Serway chap 15 -E13
Part 1: Suppose it moves as a particle under constant acceleration for a time interval of 4.5 s:
(a) Find its position for this time interval (5 points)
(b) Find its velocity at the end of this time interval (5 points)
Part 2: Next, assume it moves as a particle in simple harmonic motion for 4.5 s and x = 0 is its
equilibrium position:
(c) Find its position for this time interval (5 points)
(d) Find its velocity at the end of this time interval (5 points)
(e) Find the acceleration at half of this time interval (5 points)
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Avos
wt
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Exercise 4 (25 points)
(25 points – 2.5 pts per question): Multiple choice questions. Only one answer per question. NO
explanations are required.
1) The pressure at the bottom of a filled glass of water (ρ = 1000 kg/m3) is P. The water is poured
out, and the glass is filled with ethyl alcohol (ρ = 806 kg/m3). What is the pressure at the bottom of
the glass?
a) smaller than P
b) equal to P
c) larger than P
d) indeterminate
Explanation: Because both fluids have the same depth, the one with the smaller density (alcohol) will
exert the smaller pressure.
2) An ideal fluid flows through a horizontal pipe whose diameter varies along its length.
Measurements would indicate that the sum of the kinetic energy per unit volume and pressure at
different sections of the pipe would do what?
a) Decrease as the pipe diameter increases.
b) Increase as the pipe diameter increases.
c) Increase as the pipe diameter decreases.
d) Decrease as the pipe diameter decreases.
e) Remain the same as the pipe diameter changes.
Explanation: Since the pipe is horizontal, each part of it is at the same vertical level or has the same
y-coordinate. Thus, from Bernoulli’s equation P + ½ρv2 + ρgy = constant, we see that the sum of the
pressure remains the same.
3) A beach ball is made of thin plastic. It has been inflated with air, but the plastic is not stretched.
By swimming with fins on, you manage to take the ball from the surface of a pool to the bottom.
Once the ball is completely submerged, what happens to the buoyant force exerted on the beach ball
as you take it deeper?
a) It increases.
b) It remains constant.
c) It decreases.
d) It is impossible to determine.
Explanation: The water keeps nearly constant density as it increases in pressure with depth. The beach
ball is compressed to smaller volume as you take it deeper, so the buoyant force decreases.
4) If a simple pendulum oscillates with small amplitude and its length is doubled, what happens to
the frequency of its motion?
a) It doubles.
b) It becomes √2 times as large.
c) It becomes half as large.
d) It becomes 1/√2 times as large.
e) It remains the same.
Explanation: The period of a simple pendulum is T = 2π √(l/g), and its frequency is f =1/T=(1/2π)
√(g/l). Thus, if the length is doubled so l′ = 2l, the new frequency is f’ = (1/2π) √(g/l’) = (1/2 π) √(g/2l)
= (1/√2)(1/2 π) √(g/l) = f/√2
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5) An object–spring system moving with simple harmonic motion has an amplitude A. When the
kinetic energy of the object equals twice the potential energy stored in the spring, what is the position
x of the object?
a) A
b) 13 A
c) A/√3
d) 0
e) none of those answers
Explanation: The total energy of the object-spring system is ½ kA2 = ½mv2 + ½kx2. When the kinetic
energy is twice the potential energy, ½mv2 + 2·½kx2 = kx2 and the total energy is
½kA2 = kx2+ ½kx2 → ½kA2 = (3/2)kx2 → x = A/√3
6) A block with mass m = 0.1 kg oscillates with amplitude A = 0.1 m at the end of a spring with
force constant k = 10 N/m on a frictionless, horizontal surface. Rank the periods of the following
situations from greatest to smallest. If any periods are equal, show their equality in your ranking.
(1) The system is as described above.
(2) The system is as described in situation (1) except the amplitude is 0.2 m.
(3) The situation is as described in situation (1) except the mass is 0.2 kg.
(4) The situation is as described in situation (1) except the spring has force constant 20 N/m.
(5) A small resistive force makes the motion underdamped.
a) (1) > (3) = (4) > (2) > (5)
b) (2) > (1) > (4) > (5) > (3)
c) (3) > (5) > (1) = (2) > (4)
d) (5) = (4) > (2) > (1) > (3)
e) (3) > (2) > (5) > (1) > (4)
Explanation: The amplitude does not affect the period in simple harmonic motion; neither do
constant forces that offset the equilibrium position. Thus (1) and (2) have equal periods. The period
is proportional to the square root of mass divided by spring constant. So (3), with larger mass, has
larger period than (1). And (4) with greater stiffness has smaller period. In situation (5) the motion
is not quite simple harmonic, but has slightly smaller angular frequency and so slightly longer
period.
7) You attach a block to the bottom end of a spring hanging vertically. You slowly let the block
move down and find that it hangs at rest with the spring stretched by 15.0 cm. Next, you lift the
block back up to the initial position and release it from rest with the spring unstretched. What
maximum distance does it move down?
a) 7.5 cm
b) 15.0 cm
c) 30.0 cm
d) 60.0 cm
e) The distance cannot be determined without knowing the mass and spring constant.
Explanation: The equilibrium position is 15 cm below the starting point. The motion is symmetric
about the equilibrium position, so the two turning points are 30 cm apart.
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8) Which of the following statements is not necessarily true regarding mechanical waves?
a) They are formed by some source of disturbance.
b) They are sinusoidal in nature.
c) They carry energy.
d) They require a medium through which to propagate.
e) The wave speed depends on the properties of the medium in which they travel.
Explanation: Not all waves are sinusoidal. A sinusoidal wave is a wave of a single frequency. In
general, a wave can be a superposition of many sinusoidal waves.
9) By what factor would you have to multiply the tension in a stretched string so as to double the
wave speed? Assume the string does not stretch.
a) A factor of 8
b) A factor of 4
c) A factor of 2
d) A factor of 0.5
e) You could not change the speed by a predictable factor by changing the tension.
Explanation: From v = √(T/ µ), we must increase the tension by a factor of 4 to make v double.
10) When all the strings on a guitar (figure of the strings below) are stretched to the same tension,
will the speed of a wave along the most massive bass string be:
a) faster
b) slower
c) the same as the speed of a wave on the lighter strings
d) the speed on the bass string is not necessarily any of these answers
Explanation: Wave speed is inversely proportional to the square root of linear density.
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