Chapter 13

Dynamics

Chapter 3 Newton’s Law

NEWTON'S LAW OF INERTIA A body, not acted on by any force, remains in

uniform motion. NEWTON'S LAW OF MOTION

Moving an object with twice the mass will require twice the force.

Force is proportional to the mass of an object and to the acceleration (the change in velocity).

F=ma.

W = m*g

Fnet = F = T1 + T2 + T3 = 0

x- and y-components:T1x = - T1 cos 37o = - 0.8 T1T1y = T1 sin 37o = 0.6 T1T2x = T2 cos 53o = 0.6 T2T2y = T2 sin 53o = 0.8 T2

Solve for tension T1 and T2.Fnet,x = F x = T1 x + T2 x + T3 x = 0T1 x + T2 x + T3 x = 0- 0.8 T1 + 0.6 T2 + 0 = 0

T1 = 0.75 T2

Dynamics

M1: up as positive:Fnet = T - m1*g = m1 a1

M2: down as positive.Fnet =  F = m2*g - T = m2 a2

3. Constraint equation:a1 = a2 = a

Equations

From previous:T - m1*g = m1 a

T = m1 g + m1 a Previous for Mass 2:m2*g - T = m2 a

Insert above expr. for Tm2 g - ( m1 g + m1 a ) = m2 a

( m2 - m1 ) g = ( m1 + m2 ) a( m1 + m2 ) a = ( m2 - m1 ) g

a = ( m2 - m1 ) g / ( m1 + m2 )

Rules1. Free-Body Analysis, one for each mass + Newton’s Law

3. Algebra:Solve system of equations for all unknowns

2. Constraint equation(s): Define connections.You should have as many equations as Unknowns.COUNT!

0 = 30 0

g

i

J

m

M*g

M*g*sin

-M*g*cosj

Mass m rests on the 30 deg. Incline as shown. Step 1: Free-Body Analysis. Best approach: use coordinates tangential and normal to the path of motion as shown.

Mass m rests on the 30 deg. Incline as shown. Step 1: Free-Body Analysis.

Step 2: Apply Newton’s Law in each Direction:

0 = 30 0

g

i

J

m

M*g

M*g*sin

-M*g*cosj

xmxForces *i*sin*g*m)_(

)_(0j*cos*g*m-N )_( onlystaticyForces

N

Friction F = k*N:Another horizontal

reaction is added in negative x-direction.

0 = 30 0

g

i

J

m

M*g

M*g*sin

-M*g*cosj

xmNkxForces *i*)*sin*g*m()_(

)_(0j*cos*g*m-N )_( onlystaticyForces

N k*N

Midterm 1 :Some suggestions

•Systematic work: it takes practice, lots of it.

•Passive understanding is good, yet you still must train yourself through active practice.

•It’s all mathematical: Practice calculus and analytical geometry!

Midterm 1:Suggestions cont’d

•‘Deep Thinking 1’: Map the solution path BEFORE starting the analysis.

‘Deep Thinking 2’: Select the laws you will use.

•‘Deep Thinking 3’: Map out the Connections between the laws that will lead to the answer.

Preparing for Exam 1

•Modeling: Free-Body Analysis

•Laws and Definitions: Laws of Kinematics, Terms such as , Coriolis accel, Moving Frames of Ref.

•Seek to understand the concepts

•Practice problem solving. Copying the homework gives you an illusion and useless points

Multiple Masses A and B move in i and j-directions.

Steps: 1. Write Newton for

each Mass.2. Constraint equation

connects both masses. Here:

30 deg.

g

x

y60 deg.

vA = - vB

fig_03_007

Newton’s Law for Rotation

rramF *

amF *

Step 1: Free-Body Analysis

Step 2: Newton in radial and tangential directions

Rotation Kinematics

Similar to translation:

dt*

dt*

and dd **