Download Final Exam

Part I: Questions (Choose the correct answer)
1. A mass m is placed on a vertical spring of spring constant k compressed a distance x.
When the spring is released the mass is projected upward . The acceleration of this mass
at maximum height is
a. zero
b. g
c. –g
d.
kx
m
2. A Projectile is fired from point 0 at the edge of a cliff, with initial
velocity components of νox and νoy . The projectile rises, then
falls into the sea at point P. If the time of flight of the projectile is T.
then the horizontal distance D (see the figure) is
D
a. ν0T
b. νoyT
d. νox T
2
c. ½ g T
3. The torque exerted on the disk about an axis passing by the center is (see figure)
a. − FRiˆ
d. FRkˆ
c. FRiˆ
b. FRˆj
r
F
z
y
R
x
4. The surfaces A and B are rough and µA = 4 µB. If m A = 2m B, and both blocks come to a
stop after covering the same distance then their initial velocities are related by.
a. VA = VB
b. VA = 2VB
1
c. VA = VB
2
A
mA
B
mB
d. VA = 4VB
5. Let F be in Newton, v in meter per second, x in meter and t in seconds. The impulse is
the area under which curve?
F
vν
F
9
6
5
4
t
4
8
4
(a)
8
5
x
2
12
x x
7
2
4
6
(b)
3
6
t
9
t
2
(c)
7
10
t
(d)
Part II: Problems (solve the following problems)
1. A projectile is projected with speed vo from the ground level. It returns to the ground
after 4.0 seconds. It travels 76 meters horizontally. If air resistance is neglected, what is
the initial speed vo?
76
= 19 m s
4
1
1
2
∆y =ν οy t − g t 2 ⇒ 0 =ν οy (4) − (10 )(4)
2
2
ν ο y = 20 m / s
ν οx =
νο =
(19) + (20)
2
2
= 27.6 ≈ 28 m / s
νο
76 m
r
v v
v
2. A, B and C are three vectors of equal magnitude A=B=C= 4 units. A points to the
v
v
positive x-direction and B points to the positive y directions. Vector C is perpendicular
v
v
to vector A and makes an angle 30° with B as shown in the figure.
v v v
Find (B ΧC )⋅ A . Hint Cx= 0.
z
r
r
r
A = 4iˆ, B = 4 ˆj , C = 4 cos 30 ˆj + 4 sin 30 kˆ
r r
B × C = 8 ˆj × kˆ = 8 iˆ
r r r
B × C ⋅ A = 8(4 ) = 32
(
r
C
)
30°
x
r
B
r
A
3. Two blocks m1 =0.30 kg and m2 =0.20 kg are connected by a string, as shown In the
r
figure , and the upper block is pulled upward by a force F of magnitude 10 N. Find the
tension (in N) in the string between the two blocks
a=
F − (m1 + m2 ) g 10 − (0.3 + 0.2 )10
=
(0.3 + 0.2)
m1 + m2
a = 10 m / s
r
F
m1
2
T − m2 g = m2 a
T = m2 ( g + a ) = 0.2 (10 + 10) = 4 N
m2
4. A road way is designed for traffic moving, at a speed of 72 km/h. A curved section of
the roadway is a circular arc of 120 m radius. The roadway is banked – so that a vehicle
can go around the curve. Find the angle β (in degree) at which the roadway is banked
( ignore friction)
mν 2
N sin θ =
r
N cos θ = mg
dividing
2
 72 


2
ν
−1  3.6 
tan θ =
⇒ θ = tan
= 18°
gr
10 (120)
y
5. Two identical springs have force constants of 600 N/m. The springs are attached to a
small cube as in figure A. An external force P pulls the cube a distance x = 0.20 m to
the right and holds it there. (see fig.B) Find the work done(in J) by the external force P in
pulling the cube 0.2 m:
1

WP = 2  k x 2 
2

WP = 600 (0.2)
P
A
2
P
WP = 24 J
B
6. A block of mass 0.50 kg is compressed a distance 0.40 m against a spring of force
constant 400 N/m. When the spring is released the block moves up a frictionless
circular surface of radius 2.9m (see figure). Find the apparent weight (in N) of the block
at the top of the circular path. (point P)
E1 = E F
1 2
1
k x = mg (2r ) + mν 2 ⇒ mv 2 = kx 2 − 4 mg r
2
2
2
2
mv = (400)(0.4) − 4 (0.5)(10)(2.9)
P
mv 2 = 6
mg − N =
mv 2
r
0.5(10) − N =
r
6
2.9
x
N = 2.93 N
o
0
7. A ball m1= 0.4 kg moving with a speed of 2 m/s undergoes a head on elastic collision
with another ball m2= 1.6 kg moving in the opposite direction with a speed 3 m/s. Find
the final velocity of ball m1 (in m/s)
ν 1i −ν 2i = − (ν 1 f −ν 2 f )
Another Solution
2 + 3 =ν 2 f −ν 1 f ⇒ ν 2 f = (5 + ν 1 f
)
m1ν 1i + m2 ν 2i = m1 ν 1 f + m2 ν 2 f
(0.4)(2) + (1.6)(− 3) = 0.4ν 1 f
.8 − 4.8 = 0.4ν 1 f + 1.6 (5 +ν 1 f
− 4 = 0.4ν 1 f + 8 + 1.6ν 1 f
− 4 − 8 = 2ν 1 f
ν if = − 6 m / s
+ 1.6ν 2 f
)
ν1f =
0.4 −1.2
2(1.6)
2+
(− 3)
2
2
=− 6m / s
ν1f =
ν1f
m1 − m2
2m 2
ν 1i +
ν 2i
m1 + m2
m1 + m2
8. A uniform thin rod of length 1 m is folded at one end as shown in figure. Find the
distance of the center of mass measured from point O. (in m)
(m1 + m2 ) xCM = m1 x1 + m2 x2
(0.4 + 0.6) xCM = 0.4 (0.2) + 0.6(0.55)
O
xCM = 0.42 m
0.4 m
0.3 m
9. A rod of length 60 cm and mass of 1.2 kg rotates from rest about one end as shown in
the figure. Find the linear speed (in m/s) of the far end (A) at the lowest position.
(ICM =
1
ML2)
12
2
1
1
L
I = ML2 + M   = ML2
12
3
2
W =∆ K
L
1
MgL = I ω 2
2
2
11
ν
MgL =  ML2  2 ⇒ν = 6 gL = 6 (10)(0.6)
23
L
ν =6m / s
axis
ν?
A
10. A block m1=2.0 kg is placed on a frictionless 30o incline and connected to another
block m2 = 6 kg by a light string as shown in the figures. The pulley has a moment of
inertia 0.40 kg m2 and radius 0.2 m It rotates as the block m2 falls down. Find the
acceleration of m1. ( in m/s2)
T1 − m1 g sin θ = m1 a
m2 g − T2 = m2 a
T1
I
T2 − T1 = 2 a
R
adding
I 

m2 g − m1 g sin θ =  m1 + m2 + 2  a
R 


0.40 
6 (10) − 2 (10)(0.5) = 2 + 6 + 2  a
(2) 

a = 2.78 m / s 2
T2
m1
m2
30°