Ch 6.ppsx

Ch. 6: Work and Energy

Newton’s 2

Law is a powerful tool for

(6.1) Work

• _{work is done by}

exerting a force on an object while that object undergoes a displacement

W = F s

F

s

units of W: N m = joule or J

What if F and s are not in the same direction?

F

s

W = F cos



s

= Fs cos



Recall: scalar (dot) product AB = A B cos 

Work done by several forces

1) Calculate W done by each force, then sum to find the total work:

W_tot = W₁ + W₂ + W₃ + . . .

2) Find net force, then use:

Example 6.2

A 1500-kg sled is dragged 20 m by a tractor. The tension

in the chain is 5000 N, and it pulls at 36.9° above the horizontal. The force of friction on the sled is 3500 N.

Example 6.4

Assuming an initial speed of 2 m/s, how fast is the sled moving after it has been pulled 20 m by the tractor?

v = 4.2 m/s

(6.2) Work-Energy Theorem

m

F = ma

v

v

v₂2 = v

12 + 2as

a = (v₂2 - v 12 )

2s

= m (v₂2 - v 12 )

2s Fs = 1mv₂2 - 1mv

12

2 2

W_tot

Let’s define kinetic energy as:

K = 1mv2

(6.2) Work-Energy Theorem

W

= K

– K

=



K

Example 6.4

Assuming an initial speed of 2 m/s, how fast is the sled moving after it has been pulled 20 m by the tractor?

What factors affect stopping distance?

AP Physics Game

(maybe you can come up with a good name)

You and your opponent each get a large and small slider. You take turns sliding them across the lab

A car with mass m is traveling on a level road with speed v₀ at the instant the brakes lock, so

that the tires slide rather than roll. The coefficient of friction is _k.

a) Use the Work-Energy Theorem to calculate the minimum stopping distance.

Problem 6.27

What if _k = 0?

b) Suppose the car stops in 91.2 m if v₀ = 30 m/s. What is the stopping distance if v₀ = 15 m/s?

Problem 6.69

A 0.12-kg block is attached to a cord passing through a hole in

a frictionless, horizontal surface. The block is originally revolving at a distance of 0.40 m from the hole with a speed of 0.70 m/s. The cord is then pulled from below, shortening the radius to 0.10 m. At this new distance, the speed of the block is 2.8 m/s.

a) Find the tension in the cord before and after the cord is pulled.

(6.3) Work with Varying Forces

• _{we were forced to use the work-energy theorem}

to find the work in problem 6.69

How do you calculate the work

directly, even if the force is not

ex. A car is driving along the x-axis with the driver constantly changing his pressure on the gas pedal

x

F area of this rectangle = _{work done during this} small displacement

W = Fx

W

=



x₂

x₁

F

dx

On a graph of F vs. x, the total work done by the force is represented by the area under the curve

Problem 6.30

A child pulls a 10-kg sled along a frozen pond with force given by:

Problem 6.31

Assuming the sled was initially at rest at x = 0, find the speed of the sled at

Problem 6.66 Varying Coefficient of Friction

A box is sliding with a speed of 4.5 m/s on a horizontal surface, when, at point P, it encounters a rough section. On the rough section, the

coefficient of friction is not constant, but starts at 0.10 at point P and increases linearly with distance past P, reaching a value of 0.60 at 12.5 m past

Springs

“Hooke’s Law for springs”

F_x = kx

W =



x

0

F_x dx ₌



x

0k x dx = ½ kx

2

*In general, the work done when a spring stretches (or compresses) is given by:

W = ½ kx

- ½ kx

Problem 6.77

A 2.5-kg textbook is forced against a spring with force constant 250 N/m, compressing the spring a distance of 25 cm. When released, the book slides on a horizontal table top with coefficient of friction _k = 0.25.

Dart Gun Lab

Objective: Use the Work-Energy Theorem to calculate the spring constant of a dart gun

Rules:

Results

Problem 6.76

A 0.3-kg ball is loaded into a spring gun with force constant k = 400 N/m, compressing the spring 6 cm, the entire length of the barrel.

Problem 6.76

A 0.03-kg ball is loaded into a spring gun with force constant k = 400 N/m, compressing the spring 6 cm, the entire length of the barrel.

6 cm

a) Calculate the speed with which the ball leaves the barrel.

c) For part (b), at which position along the barrel does the ball have the greatest speed, and what is that speed?

4.5 cm

From this point on , F_spring < f so the ball starts to slow down

(6.4) Power

• _{it’s often necessary to know how quickly work}

is done

• _{power is the rate at which work is done}

P = dW_dt

• _{units of P: J/s = watt or W}

P_av = W_{ t} P_av = F_ts P_av = Fv_av