Classical MechanicsReview 2: Units 1-9

Mechanics Review 2 , Slide 1

Useful Formulas

Mechanics Review 2 , Slide 2

Work of a Force:

)( frictionexceptNCk WdfE Conservation of Energy:

Work-Kinetic Energy Theorem:

rdFWr

r

2

1

xdxFUx

xc

2

1

)(Potential Energy of a Conservative Force:

Mechanical Energy: UKE

If F is constant: rFW

Force of kinetic friction:

Force of static friction:

Potential Energy of Gravity:

Potential Energy of Spring:

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; (b) the work done on the block by the normal force; (c) the work done by friction on the block.

Mechanics Review 2 , Slide 3

q

M

0NWdmgWg )sin( q

dmgW kf )cos( q

cosrFrFW For a constant Force:

L

v

Conserve Energy from initial to final position

Example: Pendulum

Mechanics Review 2 , Slide 4

2

21 mvmgL gLv 2

mghU g 0E

L

mg

T

Example: Pendulum

Mechanics Review 2 , Slide 5

gLv 2

LmvmgT

2

mgmgmgL

mvmgT 322

2vaL L

va2

rvmmaF cy

2

Conserve Energy from initial to final position.

2v ghhh

Example: Pendulum

Mechanics Review 2 , Slide 6

2

21 mvmgh ghv 2

ghv 2

0E

mghU g

h

mgT

r

Example: Pendulum

Mechanics Review 2 , Slide 7

ghv 2

rva

2

rmvmgT

2

mgr

mghmgr

mvT 22

rvmmaF cy

2

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?

Mechanics Review 2 , Slide 8

2.5 kg

h=1.5 m

d = 0.50 m

k = 0.4

k=250 N/m

2

210)( bmvmghE a dmgkxmghdfE kk 2

21)(b

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.

Mechanics Unit 9, Slide 9

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?

Mechanics Review 2 6, Slide 10

20o

m1

m2

20o

k

qsin12 gDmgDmEi

2221 2

1)(21 kDvmmE f

01 amF 02 amFa is not constant! a = 0 when:

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) Find 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.

Mechanics Review 2 , Slide 11

00 spg UUKE

hmgU g )(21 22

ifsp xxkU

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?

Mechanics Lecture 19, Slide 12

H = 2 m

H m2

= 2 kg

m1

m2

θ

HgmHfE kk )cos( 1 qqsin12 gHmgHmEi

221 )(

21 vmmE f

All forces are constant so a is constant:

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)

Mechanics Review 2 , Slide 13

)cos1(21 2 mgRmv

RvmNmgFr

2

cos

(c) When N = 0, cosϕ0 = 2/3

gRv 2

Example: Spring and Mass

A spring is hung vertically and an object of mass m is attached to its lower end.

 1. If the spring is stretched 2.0 cm from its equilibrium position by the suspended object of mass 0.55 kg what is the spring constant k?

  2. How much work is done by the spring on the object as it stretches through this distance?

Mechanics Unit 7, Slide 14

k = 270 N/m Wsp = - 0.054 J

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?

Mechanics Review 2 , Slide 15

1kg

k

vi

2

21

ii mvmghE 2

21)( kxxmgE f

0E

00 0 mgkxmaFy

Equilibrium means a = 0: