Classical Mechanics
Review 2: Units 1-9
Second Midterm
Friday, June 17
Mechanics Review 2 , Slide 1
Important Formulas
Work of a Force:
If F is constant:
Work-Kinetic Energy Theorem:
Wnet  K
x2
Potential Energy of a Conservative Force: U    Fc ( x) d x
x1
Conservation of Energy:
Mechanical Energy:
E   f k d  WNC ( except friction)
E  K U
Forces of kinetic and static friction:
fk = mk N, fs £ ms N
Mechanics Review 2 , Slide 2
Example: Work and Inclined planes
The block shown has a mass of m = 1.58 kg. The coefficient of
friction is µk = 0.550 and θ = 20.0o. If the block is moved a
distance d = 2.25 m up the incline, calculate:
(a) the work done by gravity on the block; Wg  (mg sin q ) d
(b) the work done on the block by the normal force; WN  0
(c) the work done by friction on the block. W f    k (mg cosq ) d
(d) the change in kinetic energy W  W  W    K
net
1
2
For a constant Force:
M
q
Mechanics Review 2 , Slide 3
Example: Blocks, Pulley and Spring
Two blocks m1 = 4 kg and m2 = 6 kg are connected by a string that
passes over a pulley. m1 lies on a inclined surface of 20o with
coefficient of friction μk = 0.12, and is connected to a spring of spring
constant k = 120 N/m. The system is released from rest when the
spring is unstretched.
Find the speed of the blocks when m2 has fallen a distance D = 0.2 m.
At what distance d does the acceleration of the blocks becomes zero?
DE = - fk D
k
Ei  m2 gD  m1 gD sin q
m1
1
1 2
2
E f  (m1  m2 )v  kD
2
2
a is not constant! a = 0 when:
20o
20o
m2
 F  m1a  0  F  m2 a  0
Mechanics Review 2 6, Slide 4
Example: Block slides down the hill
A block of mass m starts from rest at the top of the frictionless,
hemispherical hill of radius R.
(a) Find an expression for the block's speed v when it is at an
angle ϕ.
(b) Find the normal force N at that angle.
(c) At what angle ϕ0 does the block "fly off" the hill?
(d) With what speed will the block hit the ground? (A) v  2 gR
E  0
1 2
mv  mgR (1  cos  )
2
v2
 Fr  mg cos   N  m
R
(c) When N = 0, cosϕ0 = 2/3
Mechanics Review 2 , Slide 5
Example: Block and spring
A 2.5 kg box is released from rest 1.5 m above the ground and
slides down a frictionless ramp. It slides across a floor that is
frictionless, except for a small section 0.50 m wide that has a
coefficient of kinetic friction of 0.40. At the left end, is a spring
with spring constant 250 N/m. The box compresses the spring,
and is accelerated back to the right.
(a) What is the speed of the box at the bottom of the ramp?
(b) What is the maximum distance the spring is compressed by
the box?
2.5 kg
k=250 N/m
h=1.5 m
k = 0.4
d = 0.50 m
1
1 2
2
(a) E  0  mgh  mvb (b) E   f k d  mgh  kx    k mg d
2
2
Mechanics Review 2 , Slide 6
Example: Pendulum
L
E  0
U g  mgh
v
Conserve Energy from initial to final position
1 2
mgL  mv
2
v  2 gL
Mechanics Review 2 , Slide 7
Example: Pendulum
T
v2
 Fy  mac  m
L
r
2
v
a
L
v2
a
L
v  2 gL
mg
mv2
T  mg 
L
mv 2
T  mg 
 mg  2mg  3mg
L
Mechanics Review 2 , Slide 8
Example: Pendulum
v v 22ghgh
hh
U g  mgh
Conserve Energy from initial to final position.
E  0
1 2
mgh  mv
2
v  2 gh
Mechanics Review 2 , Slide 9
Example: Pendulum
v  2 gh
h
r
v2
 Fy  mac  m
r
v2
a
r
T
mv2
T  mg 
r
mg
mv 2
2mgh
T
 mg 
 mg
r
r
Mechanics Review 2 , Slide 10
Example: Pulley and Two Masses
A block of mass m1 = 1 kg sits atop an inclined plane of angle
θ = 20o with coefficient of kinetic friction 0.2 and is connected
to mass m2 = 3 kg through a string that goes over a massless
frictionless pulley. The system starts at rest and mass m2 falls
through a height H = 2 m.
Use energy methods to find the velocity of mass m2 just
before it hits the ground? What is the acceleration of the
blocks?
θ
E   f k H    k (m1 g cosq ) H
Ei  m2 gH  m1 gH sin q
m2
HH
=2
m
=2
2m
kg
1
E f  (m1  m2 )v 2
2
All forces are constant
so a is constant: v 2  v02  2aH
Mechanics Lecture 19, Slide 11
Example: Two Blocks and a Pulley
A block of mass m2 on a horizontal surface with coefficient of
kinetic friction µk, is connected to a ball of mass m1 by a cord
over a frictionless pulley. A force of magnitude F at an angle θ
with the horizontal is applied to the block and the block slides
to the right.
Determine the magnitude of the acceleration of the two
objects.
å Fx,2 = F cosq - T - mk n = m2 a,
åF
åF
y,2
= n + F sin q - m2 g = 0
y,1
= T - m1g = m1a
n = -F sinq + m2 g
æ F cosq - m1g - mk n ö
a =ç
è
m1 + m2
÷
ø
Mechanics Unit 6 , Slide 12
Example: Collision with a vertical spring
A vertical spring with k = 490 N/m is standing on the ground. A
1.0 kg block is placed at h = 20 cm directly above the spring
and dropped with an initial speed vi = 5.0 m/s.
(a) What is the maximum compression x of the spring?
(b) What is the position of the equilibrium x0 of the block-spring
system?
E  0
1kg
vi
k
1 2
Ei  mgh  mvi
2 1
E f  mg ( x)  kx 2
2
Equilibrium means a = 0:
 Fy  ma  0  kx0  mg  0
Mechanics Review 2 , Slide 13
Example: Block with Friction
A 6.0 kg block, initially at rest, is pulled to the right along a
horizontal surface by a constant horizontal force F = 12 N,
applied at an angle θ = 40⁰.
Find the speed of the block after it has moved 3.0 m if the
surfaces in contact have μk = 0.15.
DE = - fk d +WNC
Mechanics Unit 9, Slide 14
Example: Popgun (Spring and Gravity)
The launching mechanism of a popgun consists of a spring.
when the spring is compressed 0.120 m, the gun when fired
vertically is able to launch a 35.0 g projectile to a maximum
height of 20.0 m above its position as it leaves the spring.
(a) Determine the spring constant (A) k = 953 N/m
(b) Find the speed of the projectile as it
moves through the equilibrium position
of the spring (A) vB = 19.7 m/s.
DE = 0 ® E f = Ei
Ug = mgh
E = K +Usp +Ug
1 2
Usp = kx
2
Mechanics Review 2 , Slide 15
Example: Spring and Mass
A spring is hung vertically and an object of mass m is
attached to its lower end. Under the action of the load mg the
spring stretches a distance d from its equilibrium position.
1. If a spring is stretched 2.0 cm by a suspended object of
mass 0.55 kg what is the spring constant k? (A) 270 N/m
2. How much work is done by the spring on the object as it
stretches through this distance? (A) -0.054 J
d
Wsp,1®2 = -k ò x dx
0
Mechanics Unit 7, Slide 16
Example: Block on Incline Plane
Suppose a block is placed on a rough surface inclined relative
to the horizontal. The incline angle is increased until the block
starts to move. Show that you can obtain μs by measuring the
critical angle θc at which this slipping just occurs.
mgsinq - fs = 0
n - mgcosq = 0
fs,max = ms n = ms mgcosqc
ms = tanqc
Mechanics Unit 6, Slide 17
Example: Hockey Puck sliding on Ice
A hockey puck on a frozen pond is given an initial speed of v0
= 20.0 m/s. If the puck always remains on the ice and slides a
distance d =115 m before coming to rest, determine the
coefficient of kinetic friction between the puck and ice.
v - v = 2da
2
2
0
åF = - f
x
k
= ma
fk = mk N = mk mg
Mechanics Unit 6, Slide 18