PRACTICE TEST 4
Allow yourself 90 minutes for this practice test.
Allowed Equations
Equations below may be used as starting equations for solving problems. If the equation you want to use
isn’t on this sheet, you’re not allowed to use it as a starting point unless the problem states otherwise. For
2-dimensional dvats, you’re expected to add the appropriate x- and y-subscripts to distinguish between
horizontal and vertical components.
vav = Δx/Δt
aav = Δv/Δt
vav = ½(vo + v)
g = 9.80 m/s2
v = vo + at
x = xo + ½(vo + v)t
x = xo + vot + ½ at 2
v 2 = vo2 + 2a( x − xo )
Fnet = ma
W = mg
fk = µkN
Wi = Fi dcosθ
K = ½ mv²
Wnet = ΔK
Ug = mgy
Ue = ½kx²
p = mv
fs <= µsN
P = W/t = Fv I = FaveΔt = Δp
ΔFres = -kΔx
ΔU = -Wc
Ptot = Σpi
a = v²/R
Wext = ΔEsys Fnet,ext Δt = ΔP
v = 2πR/T
Emech = K + U
Useful math
Values of the trigonometric functions for some common angles sin2θ + cos2θ = 1
c2 = a2 + b2
90o
0o
30o
37o
45o
53o
60o
θ
sinθ /cosθ = tanθ
1
sinθ = cos(90°- θ)
½
3/5 sqrt(2)/2 4/5 sqrt(3)/2
sinθ 0
0
sinθ = -sin(-θ)
½
cosθ 1 sqrt(3)/2 4/5 sqrt(2)/2 3/5
cosθ = cos(-θ)
1
4/3 sqrt(3) infinite
tanθ 0 sqrt(3)/3 ¾
sinθ = a/c
cosθ = b/c
tanθ = a/b
1
1. Glider 1 has a mass of 0.50 kg, and Glider 2 has a mass of 1.50 kg. At t = 0, Glider 1 is traveling
toward Glider 2 at 0.60 m/s on a horizontal, frictionless track. Glider 2 is initially stationary. The
velocity of Glider 2 after the collision is 0.20 m/s in the direction that Glider 1 was originally moving.
The collision lasts for 0.010 s.
a. If the system is composed of the two gliders only, is momentum conserved in the collision?
Explain how you know.
b. What are the magnitude and direction of the velocity of Glider 1 after the collision? Begin by
drawing a diagram of the initial and final states of the system and labeling all symbols that you
will use.
c. Was the kinetic energy of the system conserved in the collision? A calculation is required to
support your answer.
d. Determine the magnitude and direction of the average force that Glider 2 exerts on Glider 1 during
the collision.
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2. A spring fixed at its left end on a table is
compressed a distance d from its normally
relaxed position, which is at the right end of the
table shown to the right. A ball of mass m held
against the spring is quickly released and leaves
the table’s edge at horizontal velocity v. The ball
travels a horizontal distance x past the edge of
the table before hitting the floor. The height of
the table is h, and the spring constant is k.
a. Draw a diagram of the forces acting on the
ball while it is accelerating before it leaves
the table.
d
v
h
x
b. Determine an expression for the speed v in terms of k, m, and d.
c. Prove that the distance x is given by x = d
2hk
.
mg
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d. A student has a collection of spheres of the same radii but different densities and masses. The
student carries out an experiment to measure the distance x as a function of the mass m of the
sphere projected off the edge of the table. The data, which are given below, can be used to
calculate an experimental value for k. The student also measures d = 0.050 m and h = 0.750 m.
The value of g is 9.81 m/s².
m (kg)
0.0068
0.0085
0.0134
0.0309
0.0938
x (m)
2.41
2.15
1.73
1.05
0.64
What quantities should be graphed in order to yield a straight line with a slope that could be used
to determine the value of k? On the grid on the next page, plot those quantities, label the axes, and
draw the best-fit line to the data. Use the blank row(s) in the table above to record any values that
you may need to calculate.
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e. Determine the slope of the best-fit line from your graph.
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f. Use the slope of your best-fit line to calculate the experimental value of k.
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M
A
O
R
θ=60°
Spring
k
R
µk
B
d
Fixed
wall
D
C
3. In the situation shown above, a block of mass M is released from rest at point A. The block slides
without friction along a circular ramp of radius R to point B. The center of the circular path is point O.
A line drawn from O to the center of the block initially makes an angle of 60° with the vertical. From
points B to C, the block encounters a rough horizontal section of track with coefficient of kinetic
friction µk and length d. Beyond point C, the track is frictionless. At point D, the block encounters a
relaxed spring of spring constant k that is fixed at the right end to a wall. The expressions that you
determine below may be in terms of these symbols only: M, R, µk, k, d and g. Give results in reduced
form.
a. Determine the speed vB of the block at point B. Evaluate any trig functions so that there will not
be an angle in your final result.
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b. Determine how much work kinetic friction does on the block from point B to point C.
c. Determine the speed vD of the block at point D. Indicate where in your solution you use Newton’s
1st law.
d. Determine the maximum distance x that the block compresses the spring beyond its relaxed
position.
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