PHYS16 – Lecture 15

Ch. 7 Work and Energy

• Mechanical Energy– Potential Energy– Kinetic Energy

• Work– Constant Force → Dot Product– Variable Force → Integral

• Work-Mechanical Energy theorem

Ch. 7 Work and Energy

Cheat Sheet

• Mechanical Energy– Potential Energy, Gravitational– Potential Energy, Spring– Kinetic Energy

• Work– Constant Force– Variable Force

• Work-Energy Theorem

cosxFW

...dyFdxFW yx

2

21 mvK

mghU 2

21 kxU

UWKW

Work-Mechanical Energy Theorem

Work – Mechanical Energy Theorem

• Work = the transfer of Energy• Energy = the ability to do work

UWKW

Work done by system is negative

Zero of Potential Energy

• Wherever you set your zero point for potential energy will determine your coordinate system– U=0 J at y=0 point for gravitational pot. energy– U=0 J at equilibrium point for spring pot. Energy

• All calculations of ΔU will be relative so it won’t matter where you put 0 J just make sure not to switch back and forth!!!

What is the change in potential energy of this system?

http://sparknotes.com

Assume x=0 at Equilibrium point then…

J 125)5)(10(21

)(21

2

20

2

U

xxkU

What is the change in potential energy of this system?

http://sparknotes.com

Approach #1Assume x=0 at Equilibrium point then…

Approach #2Assume x=0 at ground then…

J 125)5)(10)(5()5)(10(21

)()(21

2

020

2

U

yymgxxkU

J 125)10)(10)(5()15)(10)(5(2)5)(10(U

y)15(2

)()(21

2

2

020

2

U

mgmgkxU

yymgxxkU

Accounting for zero pointnot at equilibrium

Work and spring potential energy• If mass on a spring moves, how much work is done by spring?• Use work-energy theorem to derive spring potential energy

Uxxk

Udxkx

UxdF

UW

if

x

x

f

i

)(21

)(

22

Work done by system is negative

Force and displacementare in opposite directions

Work and Kinetic energy• If an object speeds up, how much work is done on object?• Use work-energy theorem to derive kinetic energy

)(21 so...

1

)(21

)(21)(

22

22

22

if

if

if

vvmΔK

madtdvmv

vF

vvmdxdt

dtdF

vvmdxFdxd

KxdF

KEW

Assume K=mv2/2

and prove left side =right side

Just multiply and divideby dt since dt/dt=1

Now take derivative andremember to use chainrule

Energy Pre-question

• A force F is used to raise a 4-kg mass from the ground to a height of 5 m, as shown in the diagram below. What is the work done by the force F? Ignore friction and assume the rope and pulleys are massless.

A) 50 JB) 100 JC) 174 JD) 200 JE) None of the above

F

θ = 60 degrees

mass

Energy pre-question

• A projectile is shot at an angle of θ with an initial speed of v0. What is the kinetic energy of the projectile at the top of the arc?

A)

B)

C)

D)E) None of the above

0

)(cos21

)(sin2121

220

220

20

mv

mv

mv

Energy pre-question

• Some forces in nature depend on the inverse-square of the distance between objects (1/r2). If you were to calculate the potential for this force it would depend on:

A) 1/rB) 1/r3

C) rD) r2

E) None of the above

Conclusion

• Mechanical Energy– Potential Energy, Gravitational– Potential Energy, Spring– Kinetic Energy

• Work– Constant Force– Variable Force

• Work-Mechanical Energy Thereom

cosxFW

...dyFdxFW yx

2

21 mvK

mghU 2

21 kxU

UWKW

Review

1.

• A projectile is shot out by a rail gun. Assume that the projectile starts from a resting position at the right end of the gun, and that the armature applies a constant force of 3.0 N to a projectile with a mass of 0.06 kg. How long will it take for the projectile to move 1.0 m?A) 0.02 sB) 0.04 sC) 0.2 sD) 0.4 sE) None of the above.

2.

• If a projectile is shot near the surface of the Earth with a speed of 2.0 km/s horizontally, how far will it fall in 0.010 s?

A) 4.9E-4 mB) 4.9E-2 mC) 9.8E-2 mD) 2.0E1 mE) None of the above

3.

• If a projectile with a mass of 0.10 kg accelerates from a resting position to a speed of 10 m/s in 2 s, what is the force supplied by the gun to the projectile?A) 0.50 NB) 2.5 NC) 5.0 ND) 10.0 NE) None of the above

4.

• If a 0.50 kg projectile is launched straight up and is given an initial velocity of 31.6 m/s, what will be the maximum height that the projectile will rise? (Note: assume air resistance is negligible.)A) 26 mB) 51 mC) 102 mD) 204 mE) None of the above

5.

• A 7N force and an 11N force act on an object at the same time. Which of the following CANNOT be the magnitude of the sum of these forces?A) 2 NB) 8 NC) 12 ND) 18 NE) All of the values are possible

6.• A block of mass, m, is pulled across a rough floor by a rope

that exerts a force T on the block. The rope is at an angle θ with respect to the horizontal. The frictional force between the floor and the block is F. Which of the following expressions equals the frictional force F when the block moves with a constant speed?A) TB) mg-TC) T sin(θ)D) T cos(θ)E) None of the above

7.• An object is sliding at a constant speed of 10 m/s across a

level surface. If the coefficient of kinetic friction between the object and the surface is 0.7, which of the following statements correctly describes the forces acting on the object:A) Friction is the only horizontal force acting on the object.B) There is no net horizontal force acting on the object.C) There are no vertical forces acting on the object.D) The net horizontal force acting on the object causes it to accelerate.E) None of the above.

8.

• When designing his experiment to measure the acceleration of gravity, g, Galileo could have allowed spheres to drop from a height of 10 m rather than using a 10-m inclined plane. The main advantage to using the inclined plane is that on the inclined plane the:A) final velocity of a sphere is smallerB) final velocity of a sphere is largerC) spheres take longer to reach bottomD) spheres take less time to reach bottomE) final velocity of a sphere is zero

9.

• Two blocks of equal mass are connected by a mass-less horizontal rope and resting on a frictionless table. When one of the blocks is pulled away by a horizontal external force, F, what is the ratio of the net forces acting on the blocks?A) 1:1B) 1:1.41C) 1:2D) 0E) None of the above

Example of Essay Question

• List three simple tools that give a person mechanical advantage and tell how each one works. In lieu of an essay on how the tool provides a mechanical advantage you can estimate some parameters for the tool and run the numbers on what the mechanical advantage would be.