Phys 170 Lecture 23 1

Physics 170 Lecture 23

Chapter 13 - Kinetics (Dynamics)

OverviewExample problems

Phys 170 Lecture 23 2

Chapter 13: Kinetics of a Particle

Kinetics means why things move the way they do

Dynamics is another word for kinetics

Particle means we are back to points instead of rigid bodiesand we ignore rotation and moments

The forces no longer add up to zero

The net force causes acceleration

We use kinematics to describe the resulting motion.

Phys 170 Lecture 23 3

13.1: Newton’s Laws

The First Law says that if forces balance, an object stays at restor moves at constant velocity. That’s the basis for statics

The Third Law says that forces come in equal and opposite pairs. We have used that when we have two different free bodies in contact with each other.

The new one is the Second Law: if forces are imbalanced, there is an acceleration in the same direction as the net force, proportional to the net force. In equation form, .

(Newton also invented the inverse-square law of gravity).

F = ma

Phys 170 Lecture 23 4

Mass vs Weight

Weight is a force in Newtons, not a mass in kilograms.(In US units, weight is pounds, mass is slugs = 32.2 pounds).

Your weight on the Moon is 1/6 as much as on Earth, but your mass is the same.

The mass in is inertial mass.The mass in is gravitational mass.

Experimentally, all objects of any composition have the same acceleration under gravity, so inertial and gravitational mass are the same.

W = Mg

F = maF = GMm r2

Phys 170 Lecture 23 5

13.2, 13.4: Equation of Motionand Inertial Reference Frames

If there is more than 1 force, use the vector sum:

is only valid in a coordinate system that is not rotating or accelerating. This is called an inertial coordinate system.

Coordinate systems moving with constant speed in a constant direction are OK.

The rules for translating between inertial coordinate systems leave accelerations unchanged (they just change velocities).

Fi

i∑ = ma

F = ma

Phys 170 Lecture 23 6

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 7

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

N

W

F+

F−

Phys 170 Lecture 23 8

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

F+

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

a

N

W

F+

F−

Phys 170 Lecture 23 9

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

a

N

W

F+

F−

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 10

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

a

N

W

F+

F−

N − mgcosθ − masinθ = 0 + is up (and a bit left)±F + mgsinθ = macosθ + is left (and a bit down)F = µN

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 11

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 12

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθµN − macosθ = −mgsinθ

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 13

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθµN − macosθ = −mgsinθ−µmasinθ + macosθ = µmgcosθ + mgsinθ

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 14

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθµN − macosθ = −mgsinθ−µmasinθ + macosθ = µmgcosθ + mgsinθ−µ sinθ + cosθ( )a = µ cosθ + sinθ( )g

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 15

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθµN − macosθ = −mgsinθ−µmasinθ + macosθ = µmgcosθ + mgsinθ−µ sinθ + cosθ( )a = µ cosθ + sinθ( )g

a = g µ cosθ + sinθ−µ sinθ + cosθ

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 16

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθµN − macosθ = −mgsinθ−µmasinθ + macosθ = µmgcosθ + mgsinθ−µ sinθ + cosθ( )a = µ cosθ + sinθ( )g

a = g µ cosθ + sinθ−µ sinθ + cosθ

= 9.8 0.2 ⋅0.93969 + 0.34202−0.2 ⋅0.34202 + 0.93969

= 5.961 m/s2

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 17

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθµN − macosθ = −mgsinθ−µmasinθ + macosθ = µmgcosθ + mgsinθ−µ sinθ + cosθ( )a = µ cosθ + sinθ( )g

a = g µ cosθ + sinθ−µ sinθ + cosθ

= 9.8 0.2 ⋅0.93969 + 0.34202−0.2 ⋅0.34202 + 0.93969

= 5.961 m/s2

a = v2 ρ→ v = aρ = 5.961⋅100v = 24.41 m/s

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 18

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθ−µN − macosθ = −mgsinθ

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 19

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθ−µN − macosθ = −mgsinθ−µmasinθ − macosθ = µmgcosθ − mgsinθ

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 20

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθ−µN − macosθ = −mgsinθ−µmasinθ − macosθ = µmgcosθ − mgsinθ−µ sinθ − cosθ( )a = µ cosθ − sinθ( )g

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 21

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθ−µN − macosθ = −mgsinθ−µmasinθ − macosθ = µmgcosθ − mgsinθ−µ sinθ − cosθ( )a = µ cosθ − sinθ( )g

a = g µ cosθ − sinθ−µ sinθ − cosθ

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 22

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθ−µN − macosθ = −mgsinθ−µmasinθ − macosθ = µmgcosθ − mgsinθ−µ sinθ − cosθ( )a = µ cosθ − sinθ( )g

a = g µ cosθ − sinθ−µ sinθ − cosθ

= 9.8 0.2 ⋅0.93969 − 0.34202−0.2 ⋅0.34202 − 0.93969

= 1.498 m/s2

PROBLEMS 13-53 and 13-54 (page 138, 12th edition)

A 1700 kg sports car travels around a 100 m radius horizontal

circle on a 20o banked track. The coefficient of static friction

between the tires and the road is 0.20.

13-53: Determine the maximum constant speed the car can travel

without sliding up the slope.

13-54: Determine the minimum constant speed the car can travel

without sliding down the slope.

Phys 170 Lecture 23 23

a

N

W

F+

F−

N − mgcosθ − masinθ = 0±µN + mgsinθ = macosθ

µN − µmasinθ = µmgcosθ−µN − macosθ = −mgsinθ−µmasinθ − macosθ = µmgcosθ − mgsinθ−µ sinθ − cosθ( )a = µ cosθ − sinθ( )g

a = g µ cosθ − sinθ−µ sinθ − cosθ

= 9.8 0.2 ⋅0.93969 − 0.34202−0.2 ⋅0.34202 − 0.93969

= 1.498 m/s2

a = v2 ρ→ v = aρ = 1.498 ⋅100v = 12.23 m/s

Phys 170 Lecture 23 24

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 25

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

WB

F

N

TB

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 26

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

WA

WB

F

N

TB

TA

TA TA

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 27

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

WA

WB

F

N

aB aA

TB

TA

TA TA

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 28

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

TB

TA

TA TA

WA

WB

F

N

aB aA

3TA +

WA = mA

aATB +

WB +

F +N = mA

aB

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 29

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

TB

TA

TA TA

WA

WB

F

N

aB aA

3TA +

WA = mA

aATB +

WB +

F +N = mA

aB

YA :   3T − mAg = mAaA′Y B :  T − mBgsin60° − F = mBaB′X B : N − mBgcos60° = 0

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 30

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

TB

TA

TA TA

WA

WB

F

N

aB aA

YA :   3T − mAg = mAaA′Y B :  T − mBgsin60° − F = mBaB′X B : N − mBgcos60° = 0

Friction: F = µN = 0.4NPulleys:  aB = −3aA

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 31

TB

TA

TA TA

WA

WB

F

N

aB aA

3T − mAg = mAaAT − mBgsin60° − F = mBaBN − mBgcos60° = 0F = µNaB = −3aA

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 32

TB

TA

TA TA

WA

WB

F

N

aB aA

3T − mAg = mAaAT − mBgsin60° − F = mBaB → −3mBaAN − mBgcos60° = 0F = µNaB = −3aA

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 33

TB

TA

TA TA

WA

WB

F

N

aB aA

3T − mAg = mAaAT − mBgsin60° − F = mBaB → −3mBaAN − mBgcos60° = 0→ N = mBgcos60°F = µNaB = −3aA

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 34

TB

TA

TA TA

WA

WB

F

N

aB aA

3T − mAg = mAaAT − mBgsin60° − F = mBaB → −3mBaAN − mBgcos60° = 0→ N = mBgcos60°F = µN                   → F = µmBgcos60°aB = −3aA

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 35

TB

TA

TA TA

WA

WB

F

N

aB aA

3T − mAg = mAaAT − mBgsin60° − µmBgcos60° = −3mBaA

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 36

TB

TA

TA TA

WA

WB

F

N

aB aA

3T − mAg = mAaA → T = mA g + aA( ) 3T − mBgsin60° − µmBgcos60° = −3mBaA

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 37

TB

TA

TA TA

WA

WB

F

N

aB aA

3T − mAg = mAaA → T = mA g + aA( ) 3T − mBgsin60° − µmBgcos60° = −3mBaA→ mA g + aA( ) 3− mBgsin60° − µmBgcos60° = −3mBaA

Phys 170 Lecture 23 38

3T − mAg = mAaA → T = mA g + aA( ) 3T − mBgsin60° − µmBgcos60° = −3mBaA→ mA g + aA( ) 3− mBgsin60° − µmBgcos60° = −3mBaAmA

3+ 3mB

⎛⎝⎜

⎞⎠⎟ aA +

mAg3

− mBg sin60 + µ cos60( ) = 0

Phys 170 Lecture 23 39

3T − mAg = mAaA → T = mA g + aA( ) 3T − mBgsin60° − µmBgcos60° = −3mBaA→ mA g + aA( ) 3− mBgsin60° − µmBgcos60° = −3mBaAmA

3+ 3mB

⎛⎝⎜

⎞⎠⎟ aA +

mAg3

− mBg sin60 + µ cos60( ) = 0

aA = gmB sin60 + µ cos60( )− mA 3

mA 3+ 3mB

Phys 170 Lecture 23 40

3T − mAg = mAaA → T = mA g + aA( ) 3T − mBgsin60° − µmBgcos60° = −3mBaA→ mA g + aA( ) 3− mBgsin60° − µmBgcos60° = −3mBaAmA

3+ 3mB

⎛⎝⎜

⎞⎠⎟ aA +

mAg3

− mBg sin60 + µ cos60( ) = 0

aA = gmB sin60 + µ cos60( )− mA 3

mA 3+ 3mB

aA = 9.860 0.8660 + 0.4 ⋅0.5000( )−100 3

100 3+ 3 ⋅60

Phys 170 Lecture 23 41

3T − mAg = mAaA → T = mA g + aA( ) 3T − mBgsin60° − µmBgcos60° = −3mBaA→ mA g + aA( ) 3− mBgsin60° − µmBgcos60° = −3mBaAmA

3+ 3mB

⎛⎝⎜

⎞⎠⎟ aA +

mAg3

− mBg sin60 + µ cos60( ) = 0

aA = gmB sin60 + µ cos60( )− mA 3

mA 3+ 3mB

aA = 9.860 0.8660 + 0.4 ⋅0.5000( )−100 3

100 3+ 3 ⋅60= 9.8 63.96 − 33.33

213.33

Phys 170 Lecture 23 42

3T − mAg = mAaA → T = mA g + aA( ) 3T − mBgsin60° − µmBgcos60° = −3mBaA→ mA g + aA( ) 3− mBgsin60° − µmBgcos60° = −3mBaAmA

3+ 3mB

⎛⎝⎜

⎞⎠⎟ aA +

mAg3

− mBg sin60 + µ cos60( ) = 0

aA = gmB sin60 + µ cos60( )− mA 3

mA 3+ 3mB

aA = 9.860 0.8660 + 0.4 ⋅0.5000( )−100 3

100 3+ 3 ⋅60= 9.8 63.96 − 33.33

213.33aA = 1.4069m/s2

Phys 170 Lecture 23 43

3T − mAg = mAaA → T = mA g + aA( ) 3T − mBgsin60° − µmBgcos60° = −3mBaA→ mA g + aA( ) 3− mBgsin60° − µmBgcos60° = −3mBaAmA

3+ 3mB

⎛⎝⎜

⎞⎠⎟ aA +

mAg3

− mBg sin60 + µ cos60( ) = 0

aA = gmB sin60 + µ cos60( )− mA 3

mA 3+ 3mB

aA = 9.860 0.8660 + 0.4 ⋅0.5000( )−100 3

100 3+ 3 ⋅60= 9.8 63.96 − 33.33

213.33aA = 1.4069m/s2      But this is wrong!

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 44

TB

TA

TA TA

WA

WB

F

N

aB aA

aA = 1.4069 m/s2 → aΒ = −4.2209 m/s2

If B is actually moving down, the sign of kinetic friction flips!

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 45

TB

TA

TA TA

WA

WB

F

N

aB aA

aA = 1.4069 m/s2 → aΒ = −4.2209 m/s2

If B is actually moving down, the sign of kinetic friction flips!3T − mAg = mAaAT − mBgsin60° + F = mBaBN − mBgcos60° = 0F = µNaB = −3aA

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 46

TB

TA

TA TA

WA

WB

F

N

aB aA

aA = 1.4069 m/s2 → aΒ = −4.2209 m/s2

If B is actually moving down, the sign of kinetic friction flips!3T − mAg = mAaAT − mBgsin60° + F = mBaBN − mBgcos60° = 0F = µNaB = −3aA

3T − mAg = mAaAT − mBgsin60° + µmBgcos60° = −3mBaA

aA = gmB sin60 − µ cos60( )− mA 3

mA 3+ 3mB

PROBLEM 13-27 (page 126, 12th edition)

The mass of block A is 100 kg. The mass of block B is 60 kg.

The coefficient of kinetic friction between block B and the inclined

plane is 0.4. A and B are released from rest.

• Determine the acceleration of block A and the tension in the

cord. Neglect the mass of the pulleys and the cord.

Phys 170 Lecture 23 47

TB

TA

TA TA

WA

WB

F

N

aB aA

aA = gmB sin60 − µ cos60( )− mA 3

mA 3+ 3mB

aA = 9.860 0.8660 − 0.4 ⋅0.5000( )−100 3

100 3+ 3 ⋅60= 9.8 39.96 − 33.33

213.33aA = 0.3045 m/s2

T = mA g + aA( ) 3 = 336.82 N

Phys 170 Lecture 23 48

For Next Time

Good Luck on the Midterm!

Make sure you know where to go

Bring ruler etc for free-body diagrams

Bring 1 page both sides handwritten notes, no example problems, and turn it in with exam book

Bring calculator (graphing programmable OK)