2. FIGURE:
a. Mr. Peter Pettigrew pushed a 2.5 kg trash can with a 5
N force in a shiny frictionless floor. The applied force is 30
degrees from the horizontal.
30◦
5N
2.5 kg
Normal Force
3. FBD:
30◦
5N
W=2.5 kg X 9.81m/s2
Normal Force
X
Y
30◦
5N W=2.5 kg X 9.81m/s2
Normal Force
X
Y
4. 25◦
FIGURE:
b. A grocery cart is pulled by a force P which has an angle
of 25 degrees from the y-axis. The kinetic friction is 10 N.
P
W
Normal Force
10 N
6. 25◦
FIGURE:
• c. A block of mass m=3.3 kg is pushed by a distance d=2.2 m
along a frictionless table by a constant applied force of
magnitude F= 10N directed at an angle θ=25 degrees from
the horizontal.
1
F= 10 N
m= 3.3 kg
Normal Force
2
d=2.2 m
7. FBD:
W=3.3 kg X 9.81 m/s2
Normal Force
X
Y
25◦
F= 10 N
W=3.3 kg X 9.81 m/s2
Normal Force
X
Y
25◦
F= 10 N
10. FIGURE:
• e. A horizontal force of 130 N is used to push a 40-kg
lazada package at a distance of 5.00m on a rough
horizontal surface.
m= 40 kg
Normal Force
1 2
d=5.0 m
Friction
F= 130 N
13. FBD in an INCLINED SURFACE
• SAMPLE PROBLEM 1:
A car of mass m is on an icy
driveway inclined at an
angle θ as in Figure 5.11a.
(A) Find the acceleration of
the car, assuming that the
driveway is frictionless.
14. • SAMPLE PROBLEM 1:
A car of mass m is on an icy driveway
inclined at an angle θ as in Figure 5.11a.
(A) Find the acceleration of the car,
assuming that the driveway is frictionless.
𝑆𝑖𝑛𝑐𝑒 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝑜𝑛 𝑡ℎ𝑒 𝑦 − 𝑎𝑥𝑖𝑠,
FBD in an INCLINED SURFACE
15. • SAMPLE PROBLEM 2:
A ball of mass 𝑚! and a block
of mass 𝑚" are attached by a
lightweight cord that passes
over a frictionless pulley of
negligible mass, as in Figure
5.15a. The block lies on a
frictionless incline of angle θ.
(a)Find the FBD of the ball
(b)Find the FBD of the block
FBD in an INCLINED SURFACE
16. • SAMPLE PROBLEM 2:
A ball of mass 𝑚! and a block of mass 𝑚" are attached by a lightweight cord that passes over a
frictionless pulley of negligible mass, as in Figure 5.15a. The block lies on a frictionless incline of angle θ.
(a) Find the FBD of the ball
(b) Find the FBD of the block
(a) FBD of the ball
(b) FBD of the block
FBD in an INCLINED SURFACE
17. WORK DONE BY A SPRING
• If the spring is either stretched or compressed a small
distance from its un stretched (equilibrium) configuration,
it exerts on the block a force that can be expressed as:
• where x is the position of the block relative to its
equilibrium (x ≠ 0) position and k is a positive constant
called the force constant or the spring constant of the
spring.
𝐹! = −𝑘𝑥
21. Sample Problem 2
When a 4.00-kg object is hung vertically on a certain
light spring that obeys Hooke’s law, the spring
stretches 2.50 cm. If the 4.00-kg object is removed,
(a) how far will the spring stretch if a 1.50-kg block is
hung on it, and (b) how much work must an external
agent do to stretch the same spring 4.00 cm from its
unstretched position?
𝑮𝒊𝒗𝒆𝒏:
𝑥&'( = 2.5 𝑐𝑚 𝑜𝑟 2.5 𝑥 10#"
𝑚 𝑥!.*'( =?
𝑚 = 0.55 𝑘𝑔
𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:
𝑊
$@ 4 𝑐𝑚 =?
𝐴. )𝑥!.*'( =?
𝑘 =
𝑚𝑔
𝑥
𝑘 =
(4)(9.81)
(2.5 𝑥 10#")
𝑘 =
(4)(9.81)
(2.5 𝑥 10#")
𝒌 = 𝟏𝟓𝟔𝟗. 𝟔
𝑵
𝒎
𝑥!.*'( =
(1.5)(9.81)
(1569.6)
𝒙𝟏.𝟓𝒌𝒈 = 𝟗. 𝟑𝟕𝟓 𝒙 𝟏𝟎X𝟑𝒎 𝒐𝒓 𝟎. 𝟗𝟑𝟖 𝒄𝒎
22. Sample Problem 2
When a 4.00-kg object is hung vertically on a certain
light spring that obeys Hooke’s law, the spring
stretches 2.50 cm. If the 4.00-kg object is removed,
(a) how far will the spring stretch if a 1.50-kg block is
hung on it, and (b) how much work must an external
agent do to stretch the same spring 4.00 cm from its
unstretched position?
𝑮𝒊𝒗𝒆𝒏:
𝑥&'( = 2.5 𝑐𝑚 𝑜𝑟 2.5 𝑥 10#"
𝑚 𝑥!.*'( =?
𝑚 = 0.55 𝑘𝑔
𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:
𝑊
$@ 4 𝑐𝑚 =?
𝐵. )𝑊
$@ 4 𝑐𝑚=?
𝑊
$ =
1
2
𝑘𝑥+
"
−
1
2
𝑘𝑥%
"
𝑾𝒔 = −𝟏. 𝟐𝟓𝟔 𝑱
𝑊
$ =
1
2
(0) −
1
2
𝑘𝑥%
"
𝑊
$ = −
1
2
(1569.6)(4𝑥10#")"
24. WORK DONE BY VARYING FORCE
∑𝑊 = 𝑊[] = #
3!
3"
∑𝐹3 𝑑𝑥 𝑾 = $
𝒙𝒊
𝒙𝒇
𝑭𝒙 𝒅𝒙
25. SAMPLE PROBLEM 1:
A force acting on a particle varies with x, as
shown in the figure. Calculate the work done
by the force as the particle moves from 𝑥 =
0 to 𝑥 = 6.0 𝑚.
2
1
𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:
𝑇𝑜𝑡𝑎𝑙 𝑊𝑜𝑟𝑘 = 𝐴𝑟𝑒𝑎 1 + 𝐴𝑟𝑒𝑎2
𝐴𝑟𝑒𝑎1 = 4𝑚(5𝑁) = 20 𝑁 − 𝑚
𝐴𝑟𝑒𝑎2 =
1
2
[ 2𝑚 5𝑁 ] = 5 𝑁 − 𝑚
𝑇𝑜𝑡𝑎𝑙 𝑊𝑜𝑟𝑘 = 𝐴𝑟𝑒𝑎 1 + 𝐴𝑟𝑒𝑎2
𝑇𝑜𝑡𝑎𝑙 𝑊𝑜𝑟𝑘 = 20 + 5 𝑁 − 𝑚
𝑇𝑜𝑡𝑎𝑙 𝑊𝑜𝑟𝑘 = 25 𝑁 − 𝑚 𝑜𝑟 25 𝐽𝑜𝑢𝑙𝑒𝑠
26. SAMPLE PROBLEM 2:
The interplanetary probe shown in Figure 7.9a
is attracted to the Sun by a force given by:
in SI units, where x is the Sun-probe separation
distance. Graphically and analytically
determine how much work is done by the Sun
on the probe as the probe–Sun separation
changes from 1.5𝑥10!!𝑚 to 2.3𝑥10!!𝑚.
𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:
𝑊 = #
3!
3"
𝐹3 𝑑𝑥 = #
^._3^`##
5.a3^`##
−1.3𝑥1055
𝑥5 𝑑𝑥
27. SAMPLE PROBLEM 2:
The interplanetary probe shown in Figure 7.9a
is attracted to the Sun by a force given by:
in SI units, where x is the Sun-probe separation
distance. Graphically and analytically
determine how much work is done by the Sun
on the probe as the probe–Sun separation
changes from 1.5𝑥10!!𝑚 to 2.3𝑥10!!𝑚.
𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏:
𝑊 = #
3!
3"
𝐹3 𝑑𝑥 = #
^._3^`##
5.a3^`##
−1.3𝑥1055
𝑥5 𝑑𝑥
𝑊 = −1.3𝑥1055 #
^._3^`##
5.a3^`##
𝑥X5 𝑑𝑥
= −1.3𝑥1055(
𝑥X^
−1
) ∣^._3^`##
5.a3^`##
= −1.3𝑥1055(
−1
2.3𝑥10^^ −
−1
1.5𝑥10^^)
𝑊 = −3𝑥10KL
𝐽𝑜𝑢𝑙𝑒𝑠
