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1. Classical Mechanics
- A person standing on the surface of the Earth throws a ball. The ball leaves the thrower’s hand with initial velocity vi and has final velocity vf just before it is caught. If air resistance is negligible, which of the following diagrams correctly represents a possible sequence of velocity vectors for the ball?
Vertical
Horizontal
Vertical
Horizontal
Vertical
Horizontal
Vertical
Horizontal
Vertical
Horizontal
( A )
( B )
( C )
( D )
( E )
m
h
- As shown above, a block of mass m is released from rest at a distance h above a vertical massless spring with spring constant k, what is the maximum kinetic energy of the block? (A) mgh
(B)
k
m g
mgh
2 2
(C)
k
mgh m^2 g^2
+^1
(D)
k
mgh + m^2 g^2
(E) 2mgh
m v M
vB = 0
Before After
- A bullet of mass m traveling at speed v strikes a block of mass M, initially at rest, and is embedded in it as shown above. How far will the block with the bullet embedded in it slide on a rough horizontal surface of coefficient of kinetic friction μk before it comes to rest?
(A) )
2
g
v
m
m M
μ k
(B) )
2
g
v
M
m M
μ k
(C) )
2 2
g
v
M
m M
μ k
(E) 2 2
1
M ( R +^1 r ω
- If the string is pulled slowly downward so that the center of the puck moves in a circle of smaller radius, quantities that are conserved include which of the following? I. Angular momentum II. Linear momentum III. Kinetic energy (A) I only (B) III only (C) I and II only (D) II and III only (E) I, II, and III
- An expression for the potential energy of two ions is
9
2
r
b
r
kq
P E +
What is the constant b as a function of the equilibium spacing r 0?
(A) (^10) 0
2
r
kq
(B)
0
2
r
kq
(C)
0
2
r
kq
(D)
0
kq^2 r
(E)
8 0
kq^2 r
- A rigid cylinder rolls at constant speed without slipping on top of a horizontal plane surface. The acceleration of a point on the circumference of the cylinder at moment when the point touches the plane is (A) directed forward (B) directed backward
(C) directed up (D) directed down (E) zero
Questions 9- A cylinder with moment of inertia 4 kg⋅m^2 about a fixed axis initially rotates at 80 radians per second about this axis. A constant torque as applied to slow it down to 40 radius per second.
- The kinetic energy lost by the cylinder is (A) 80 J (B) 800 J (C) 4000 J (D) 9600 J (E) 19,200 J
- If the cylinder takes 10 seconds to reach 40 radius per second, the magnitude of the applied torque is (A) 80 N⋅m (B) 40 N⋅m (C) 32 N⋅m (D) 16 N⋅m (E) 8 N⋅m
Questions 11- A nonrelativistic particle of mass m moves in a plane. Its position is described by the polar
coordinates r and θ, with time derivatives r & and θ&^. There exists a potential energy U = kr^2 ,
where k is a constant.
- Which of the following is the Lagrangian of the particle?
(A) ( 2 2 2 ) 2
L = mr & + r θ& − kr
(B) ( 2 2 ) 2
L =^1 mr & +θ + kr
(C) ( 2 2 2 2 ) 2
L =^1 m θ r & + r θ& − kr
(D) ( 2 2 2 ) 2
L =^1 mr & + rr &θ& + r θ& − kr
(E) ( 2 2 2 2 ) 2
L = mr & + rr &θ& + r θ& − kr
- What is the number of degrees of freedom for 6 particles moving freely in one plane? 6 (B) 8
ass m 1 at the end of a spring executes simple harmonic motion with a period T 1. The eriod of oscillation of a different mass m 2 on the same spring is
(A)
(C) 10
(D) 12
(E) 18
- A m p
1
1 2
m
T m
(A)
2
1
1 m
m
(B) T
(C) T 1
(D)
2
m
m
π
(E)
1
m
m
π
- Under the influence of a mutual interaction, an object orbits another object that is fixed. The rbit lies in a plane and the areas swept out by the radius vector in equal times are equal. What can e correctly concluded about the force between the objects?
nal. sions is justified.
o b (A) It is central. (B) It is inverse-square. (C) It is conservative. (D) It is gravitatio (E) None of these conclu
Questions 18-
l
l'
In two experiments, two cylinders, X and Y, are released from rest at the top of the same inclined ramp and roll down without slipping. Let tX and tY be the respective times taken for the cylinders to reach a particular line ll’ on the ramp shown above.
at of Y, MX = MY. Which of the llowing relationships between tX and tY is correct?
tX
ond experiment, both cylinders are solid, uniform, of identical length, and of the me density; but their radii are different so that the mass of X is twice that of Y, M (^) X = 2MY. hich of the following relationships between tX and tY is correct?
tX
l is hit from ground level with an initial velocity v 0 at an angle θ with respect to the round. If air resistance is negligible and the magnitude of the gravitational acceleration is g, the all hits the ground at what distance from the point at which it was hit?
- In the first experiment, both cylinders are solid, uniform, and of identical dimensions; but they are made of different materials so that the mass of X is twice th fo (A) tX ≥ 2tY (B) 2tY > tX > tY (C) tX = tY (D) 2tX > tY > (E) tY ≥ 2tX
- In the sec sa W (A) tX ≥ 2tY (B) 2tY > tX > tY (C) tX = tY (D) 2tX > tY > (E) tY ≥ 2tX
- A golf bal g b
(A)
g
v 02
(B)
g
v 02 sin θ
(C)
g
v 02 cos θ
2 3. A particle moves in a force field given by F = r^2 r , where r is the position vector. If there are o other forces, quantities that remain constant include which of the following?
mentum about the origin
only only
n I. Total energy II. Torque about the origin III. Angular mo (A) I only (B) III only (C) I and II (D) II and III (E) I, II, and III
v 0 m 3 m 2 k m 1
- Particle 1 of mass m 1 and particle 2 of mass (^2 )
m = 1 m are coupled by a massless spring of
force constant k and lie at rest on a horizontal frictionless sur face as shown above. A third particle
of mass (^3 )
m = m = m and speed v 0 strikes particle 2 along the axis of the spring and sticks
to particle e center of mass of the system after the collision is
(A)
- The speed of th
v 0
(B) v^0
(C)
v 0
(D) v 0
1
(E) v 03
m
k
- A planet of mass m moves about the Sun of mass M. G is Newton’s constant, r is the planet’s istance from the Sun, and v is the planet’s speed. Except for an additive constant the planet’s
d potential energy is
(A)
r
mv^2 +^ GMm
r
mv^2 −^ GMm
(B)^1
r
GMm
(C) −
r^2
(D) −^ GMm
(E) mgr
Q
- The diagram above show a top view of a phonograph turntable mounted on a frictionless bearing. Initially, the turntable is at rest and a massive bug is asleep at point Q. The bug wakes up,
ion only if the walk did not encircle the pivot point same position whether or not the walk encircled the pivot
(E) cessarily at rest, whether or not the pivot point was encircled
takes a walk such as the one indicated by the dotted line, returns to Q, and goes back to sleep. Afterward, the turntable is (A) again at rest in the same position (B) at rest, but in the same posit (C) at rest, but not necessarily in the point (D) at rest only if the walk did not encircle the pivot point not ne
2 m
m
m
2 m
U
L
- The weight of a door is entirely supported by the upper hinge U which is 2 meters from the lower hinge L as shown above. Assume the mass per unit area of the door is constant and the hinges have negligible size. If the door weighs 200 newtons and is 2 meters wide, of what magnitude is the horizontal force exerted by the lower hinge L? (A) 80 N (B) 100 N
v 1 v 2 v 1
p 1 p 2
- Water flows through a Venturi tube as shown in the diagram above. The radius of the large cross section of the pipe is 2 centimeters and the raduis of the constricted portion of the pipe is 1 centimeter. If the speed of the water in the large cross section is 1 meter per second, the pressure difference (p 1 − p 2 ) is most nearly (A) 0.6× 102 N/m (B) 3 × 102 N/m (C) 1.5× 103 N/m (D) 7.5× 103 N/m (E) 37.5× 103 N/m
Answer Sheet
No. Answer C or I 1 E 2 C 3 E 4 B 5 E 6 A 7 E 8 C 9 D 10 D 11 A 12 E 13 D 14 D 15 D 16 A 17 A 18 C 19 C 20 E 21 E 22 E 23 E 24 A 25 C 26 C 27 B 28 B 29 C 30 D 31 B
(C)
2
1
R
qR
(D)
1
2
R
qR
(E) )
R 1 R 2
q +
e 1 e 2
- In the circuit illustrated above, the point marked e 1 is raised instaneously at time t 0 from zero to a fixed positive potential. If e 2 is connected to a high-impedance D.C. oscilloscope input, the gragh of its voltage V as a function of time t will most resemble which of the following?
t t
t t
V V
O O
O O
V V
V
O t
t 0 t (^0)
t 0 t (^0)
t (^0)
( A ) ( B )
( C ) ( D )
( E )
b
c
i
i
r P
a
- a coaxial cable having radii a, b, andc carries equal and opposite currents of magnitude i on the inner and outer conductors.What is the magnitude of the magnetic induction at point P outside of the cable at a distance r from the axis? (A) Zero
(B) (^02)
2 a
ir
π
μ
(C)
r
i
π
μ
0
(D) 22
2 2 0
2 c b
c r
r
i
π
μ
(E) 22
2 2 0
2 c b
r b
r
i
π
μ
A
B
C (^) D E
A, B, C, D, E
r
Electric Field Magnitude
O R
- A isolated sphere of radius R contains a uniform volume distribution of positive charge. Which of the curves on the graph above correctly illustrates the dependence of the magnitude of the electric field of the sphere as function of the distance r from its center?
(B) 0.6 V
(C) 1.2 V
(D) 1.5 V
(E) 3.0 V
- A conducting cavity is driven as an electromagnetic resonator. If perfect conductivity is assumed, the transverse and normal field components must obey which of the following conditions at the inner cavity walls? (A) En = 0, Bn = 0 (B) En = 0, Bt = 0 (C) Et = 0, Bt = 0 (D) Et = 0, Bn = 0 (E) None of the above
A
B
- A uniform and constant magnetic field B is directed perpendicularly into the plane of the page everywhere within a rectangular region as shown above. A wire circuit in the shape of a semicircle is rotated counterclockwise in the plane of the page about an axis A. The axis A is perpendicular to the page at the edge of the field and directed through the center of the straight-line portion of the circuit. Which of the following graphs best approximates the emf ε induced in the circuit as a function of time t?
O t
( A )^ ε
O t
( B )^ ε
O t
( C )^ ε
O t
( D )^ ε
O t
( E )^ ε
X
Y
P
Q
- The diagram above represents the electric field and the equipotential line for two charged conductors, X and Y. The charge on X is positive. Which of the following statements is NOT correct? (A) The charges on X and Y are of the same sign. (B) The charge on X is greater than that on Y. (C) The electric field lines and the equipotential lines always intersect each other at an angle of 90 degrees. (D) A test charge placed at point P would be in equilibrium under the action of the electrical forces. (E) The electrical potential at point P is less than it is at point Q, which is symmetrically located on the other side of X.
- The potential difference across the terminals of a battery is 10 volts if no current is drawn from it. When a 4-ohm resistor is connected across the terminals, a current of 2 amperes is drawn from the battery. This resistor is then removed and replaced with a variable resistor that can have any value of resistance from 0 to ∞. If the internal resistance of the battery is constant, the maximum power P the battery can deliver to the variable resistor is (A) 16 Watts (B) 25 Watts (C) 50 Watts (D) 100 Watts (E) arbitrarily large as the resistance goes to zero
