Relationship between Force and Potential Energy

Energy Diagrams for Mechanical SystemsEnergy Diagrams for Mechanical Systems

EEtot

U(y) = mgyEtot = U(y1) + KE(y1)

ydUFdy

mg KE(y1)

Note: since U = U(y), we can (y)determine all of the components of F:U(y1) F U

xi U

yj U

zk mgj

(y1)

yy1

Energy Diagrams for Mechanical SystemsEnergy Diagrams for Mechanical Systems

E U(y)EtotU(y)

Etot = U(y1) + KE(y1)

½ mv2At this point, Etot = U(y), so KE 0 0KE = 0 v=0; maximum excursion in y

mgy1

excursion in y. Motion is bounded in y.gy1

yy1

SpringsSprings

E U(x) = ½ kx2

EEtot

dUF kxd

KE(x1) = ½ mv2

dx

U(x1) = ½ kx12

x2x1

2

2 0d Udx

Stable Equilibrium; Force always opposite displacement

Stable Equilibrium?Stable Equilibrium?

E

dU/dx > 0 dU/dx < 0

F < 0 F > 0F < 0 F > 0

xxd2U/dx2 < 0 everywhere Unstable Equilibrium

Arbitrary Energy DiagramsArbitrary Energy Diagrams

E

U(x)U(x)

x

QuantizationQuantization

EE4E4

E33

E22

E1

E0

x

Quantization and Atomic Effects

The Work-Energy Theorem

A 2-kg block slides down a frictionless curved ramp, starting from rest at a height of 3 m. The block then slides 9 m on a rough horizontal surface before coming to rest.( ) Wh t i th d f th bl k t th b tt f th ?(a) What is the speed of the block at the bottom of the ramp?(b) What is the energy dissipated by friction?( ) Wh i h ffi i f ki i f i i b h bl k(c) What is the coefficient of kinetic friction between the block and the horizontal surface, assuming that we can ignore any energy that goes into heating the block?energy that goes into heating the block?

A mass m attached to a spring of spring constant k executes uniform circular motion on a frictionless horizontal table. The spring has an unstretched length of L0, and the radius of the circle of motion is Rof motion is R.

( ) Fi d th t ti l t d i th i(a) Find the potential energy stored in the spring (b) Find the kinetic energy of the mass ( ) N f i i i d b h d h(c) Now suppose friction is turned on between the mass and the table, and that the coefficients of kinetic and static friction are equal and small << 1 How far does the mass move before itequal and small, << 1. How far does the mass move before it stops?

Two blocks are attached by a massless string over a massless pulley. The coefficient of kinetic friction between the block and the horizontal surface is k = 0.4. The system starts from rest. The block on the surface has a mass of 3 0 kg and the hanging blockblock on the surface has a mass of 3.0 kg and the hanging block has a mass of 5.0 kg. What is the speed of the hanging block after it has fallen 1.5 m? Assume the sizes of the blocks are negligible.g g

A child of mass M slides down a frictionless hemisphere of radius R. a) If the child starts from rest at the top, at what height above the ground does she lose contact with the hemisphere?b) If we now introduce friction into the problem, and the coefficient of static friction between the child and the sphere is 0.1, how far down the sphere must she sit before she starts to slide?down the sphere must she sit before she starts to slide?c) The child moves to this point, then starts to slide. Is the height at which she flies off less than or greater than in part a) ?which she flies off less than or greater than in part a) ?

R

A straight rod of negligible mass is mounted on a frictionless pivot, as shown below. Blocks have masses m1 and m2 are attached to the rod at distances 1 and 2. (a) Write an expression for the gravitational potential energy of the blocks as a functionfor the gravitational potential energy of the blocks as a function of the angle made by the rod and the horizontal. (b) For what angle is this potential energy a minimum?g p gy

A pendulum bob of mass m is attached to a light string of length L and is also attached to a spring of force constant k. With the pendulum in the position shown in the figure, the spring is at its unstressed length If the bob is now pulled aside so that the stringunstressed length. If the bob is now pulled aside so that the string makes a small angle with the vertical and released, what is the speed of the bob as it passes through the equilibrium position? p p g q pHint: Recall the small-angle approximations: if is expressed in radians, and if <<1, then sin tan and cos 1 –½ 2.

Power

Force A does 5 J of work in 10 sec. Force B does 3 J of work in 5 sec. Which force delivers greater power?

A single force of 5 N acts in the x-direction on an 8 kg object. (a) If the object starts at rest at x = 0 at time t = 0, find its ( ) e objec s s es e , d s

velocity v as a function of time. (b) Write an expression for the power input as a function(b) Write an expression for the power input as a function

of time. (c) What is the power input of the force at time t 3 sec?(c) What is the power input of the force at time t = 3 sec?

A sports car accelerates from zero to 30 mph in 1.5 s. How long does it take for it to accelerate from zero to 60 mph, assuming the power of the engine to be independent of velocity and neglecting friction?

1. 2 s 2 3 s2. 3 s 3. 4.5 s 4. 6 s 5. 9 s 6. 12 s

What does v(t) look like for constant power?