Lesson 5: Work and Energy

*CRB* Say that Arnold Schwarzenegger's biceps have the power of 500W. How long would he have to contract his muscles to do 1250J of work? (A) 2.5 seconds (B) 7.5 seconds (C) 2.5 minutes (D) 7.5 minutes

(A) 2.5 seconds Power = work/time Work = Power x Time 1250= (500)(Time) Time = 1250/500 = 2.5 seconds

If a box (9.88 kg) is moving at 11.32 m/s and is slowed by friction until the velocity is 4.89 m/s, what is the work done by friction (in J)? (A) -751 (B) -515 (C) 515 (D) 751

(B) -515 Work for moving object = change in kinetic energy KE = 1/2 mv^2 ΔKE = 1/2 (9.88)(4.89)^2 - 1/2 (9.88)(11.32)^2 ΔKE = (approx. 125 (actual: 118)) - (approx. 600 (actual: 633)) Work done = approx. -475 (actual: -515) This can also be calculated using the vf^2 kinematic formula

If a spring is exerting a force of 7.23 N at a displacement of -9.65 m, what is the spring constant for that spring (in N/m)? (A) .642 (B) .749 (C) .813 (D) .893

(B) .749 Force = 7 N Displacement = approx. -10 m (actual -9.65 m) F = -kx 7.23 = -k (-9.65) k = 7.23/ 9.65 k = approx. .75 (actual: .749) So the spring had a force of -7.23 N done on it the higher the value of K the more stiff the spring, and the lower the value of k the less stiff the spring is

Johnny does 209.87 J of work to a box of mass 4.04 kg that was sitting still on frictionless ice. What is the velocity of the box (in m/s)? (A) 5.67 (B) 10.19 (C) 16.78 (D) 24.67

(B) 10.19 Work = approx. 210 J In this example, all of that work is converted to kinetic energy because there is no energy lost from friction, so KE = approx 210 J (KE shows the amount of work that needed to be put in to accelerate object to a new velocity) KE = 1/2 mv^2 209.87 = 4.04/2 V^2 approx. 100 (actual: 103.9) = v^2 v = approx. 10 m/s (actual: 10.19) also know how to do this equation using kinematic formula

How much power does it take to lift a 96.57 kg weight 1.34 m into the air in 7.25 seconds? (A) 97.65 (B) 174.92 (C) 302.65 (D) 425.43

(B) 174.92 Work Done = PE = mgh PE = (96.57)(9.8)(1.34) = approx. 1000 J (actual: 1268.16) Power = work/time Power = 1268.16 J / 7.25 s Power = approx. 200 J/s (actual: 174.92) The more power output is the faster work can be done on an object

If Johnny pushes a box with a force of 21.24 N over a distance of 10.76 m, how much work has Johnny done to the box (in J)? (A) 165.9 (B) 228.5 (C) 302.6 (D) 452.1

(B) 228.5 Work = Force x Distance Work = 21.24 N x 10.76 m Work = approx. 200 (actual: 228.5 Nm) Note that one N*m is equal to 1 Joule

*CRB* In which of the following scenarios would you be able to calculate the appropriate variable? I. Given Gravitational Potential Energy and Elastic Potential Energy, find Kinetic Energy. II. Given Kinetic Energy and Mechanical Energy, find Elastic potential Energy. III. Given total Potential Energy and Mechanical Energy, find Kinetic Energy. (A) I only (B) III only (C) I and II only (D) I and III only

(B) III only In the following scenario, you would be able to find the asked-for variable: Given total Potential Energy and Mechanical Energy, find Kinetic Energy. I. You have no Mechanical Energy. II. You have no way to differentiate between Elastic and other forms of Potential Energy

If a box has a mass of 5.34 kg, and is pushed by a spring for 4.89 m, giving it a velocity of 11.29 m/s, what is the spring constant for that spring (in N/m)? (A) 14.5 (B) 21.5 (C) 28.5 (D) 43.5

(C) 28.47 KE = 1/2 mv^2 KE = (.5)(5.34)(11.29)^2 KE = approx. 350 J (actual: 340) I found KE using W=Fd and then used the vf^2 kinematic formula to find the acceleration when we were given no rime KE = PE = 1/2 kx^2 340 J = k(.5)(4.89)^2 k = approx. 30 kg/s^2 (actual: 28.5) Since the box was not moving before hand, the amount of work done on the spring by the box is equal to the amount of potential energy stored in the spring. Newtons third law. The potential energy of the spring is what caused the box to move. Conservation of energy assuming no friction Based on the amount of energy that was but into the box based on the work done by the siring, we knew to keep the spring stable, the amount of work done by the box on the spring was equal *draw out in notebook* Note: kg/s^2 is equivalent to N/m

If a car is moving 28.76 m/s and the engine is applying a force of 114.89 N, what is the power output of the engine at that moment (in W)? (A) 2164.32 (B) 2597.65 (C) 3304.24 (D) 4587.92

(C) 3304.24 Power = Force x Velocity Power = (114.89)(28.76) Power = approx. 3000 J/s (actual: 3304.24)

If a 18.77 N box is sitting on one end of a lever at a distance of 1.32 m from the fulcrum, how much force must be applied to at a distance of 4.35 m from the fulcrum to raise the box? (A) 3.43 (B) 4.89 (C) 5.70 (D) 7.42

(C) 5.70 f1 x d1 = f2 x d2 18.77 x 1.32 = 4.35 x f2 f2 = approx. 6 N (actual: 5.70)

If a 9.76 kg ball is raised to a height of 10.34 m, what is the potential energy of the ball (in J)? (A) 675.8 (B) 865.3 (C) 989.0 (D) 1143.6

(C) 989.0 W=PE = mgh PE = (9.76)(9.8)(10.34) PE = approx. 1000 (989.00) Ball is stationary and moved a VERTICAL distance thus it only has PE

*CRB* There is also a Conservation of Momentum, based on the fact that the force exerted by one object onto the second is equal and opposite to the force exerted by the second object on the first. Which of Newton's Laws is this based upon? (A) First Law (B) Second Law (C) Third Law (D) I will go review Newton's Laws and come back to this!

(C) Third Law Newton's Third Law is based on equal and opposite reactions, so this should make sense.

*CRB* In the previous card's second example, the altitude of the object wasn't changing, but rather its reference point as the ground (or 0 potential energy position). What is the proper term for this reference point? (A) Absolute Height (B) Relative Height (C) Base (D) Datum

(D) Datum The Datum is the 0 potential energy position, often thought of as the ground.

*CRB* In which of the following scenarios would the gravitational potential energy increase? I. Increasing the height of the object over the flat ground. II. Keeping the object at the same altitude, but the ground is much lower (like hovering over a cliff instead of flat ground). III. Working on a different planet with a larger force of gravity. (A) I only (B) I and II only (C) I and III only (D) I, II and III

(D) I, II and III Each of the following scenarios would increase Gravitational Potential Energy: I. Increasing the height of the object over the flat ground. II. Keeping the object at the same altitude, but the ground is much lower (like hovering over a cliff instead of flat ground). III. Working on a different planet with a larger force of gravity.

How to calculate how much work was done to compress or stretch a spring?

1/2 k x^2 This tells you how much work was done on the spring, and the negative of this is how much work spring is doing on you. This work formula will also tell you how much potential energy is stored in the spring was stretched or compressed

What can be said about conservative forces and closed pathways?

A force is conservative if the net work it does on an object when the object moves in a closed pathway is zero

*CRB* If you are told the total work that Friction or Air Resistance has on an object, would you be able to trace its path? Why or why not?

Although these are non-conservative forces and we know that the total work done is dependent on the path, simply knowing the amount of work done is not sufficient to determine what path the object took.

What is the difference between work and moment when dealing with mechanical advantage?

Both work and moment are equal to distance times force, but with work the force is in the same direction as the distance, with moment the distance is perpendicular.

Conservation of energy problem: A 90 kg object start at rest from a 500 m long hill with a 5 degree incline. Assuming average force of friction of 60 N. Find speed of biker at the bottom of the hill

Conservation of energy E1+ E2+...= Ef Note that Work in this case is seen as energy (ex. PE= mgh but also W=mgh or F*d= mgh) Work done in opposite direction of displacement is (-) Rider at top of hill is above some zero point so the rider must have gravitational potential energy At the bottom of the hill, that rider is going to have KE because it started at rest (KE of 0) and now its velocity has obviously increased along the way causing it to accelerate, but note that not all of the PE is converted to KE because of friction acting against the object. Friction is a nonconservative force that "eats up" mechanical energy because when you these types of forces at play, all of the force is not conserved Final velocity is 13.7 m/s

Is Gravitational Force a conservative or non-conservative force? Friction? Force of a Spring? Air Resistance?

Conservative: Gravitational force, Force of a Spring Non-conservative: Friction, Air Resistance

*CRB* What is the equation for Total Mechanical Energy E?

E = KE + PE KE- Kinetic Energy PE- Potential Energy

What is the equation for Hooke's law (restorative force of a spring in terms of displacement)?

F = -kΔx F = Restorative force k = spring constant x = displacement Restorative force almost behaves like Normal force. If the spring is no longer accelerating in a certain direction and has stopped the force acting against the force you are placing on it is equal to it. Restorative force is always in the opposite direction of how much you displace it the spring. So restorative force is the force that is trying to get the spring back to its normal position

*CRB* True or false? All changes in potential energy are equal to the opposite of work done by gravity (PE = -Wgrav).

False. Changes in potential energy due to gravity are equal to the opposite of work done by gravity (PEgrav = -Wgrav). Recall there are other types of potential energies

*CRB* True or False? Conservation of Total Mechanical Energy claims that any forces acting on objects will not affect the total Mechanical Energy of the system.

False. Conservation of Total Mechanical Energy claims that any Conservative Forces acting on objects will not affect the total Mechanical Energy of the system.

*CRB* Why couldn't Non-Conservative Forces be assigned any arbitrary paths and be treated independently like Conservative Forces?

For Non-Conservative Forces, there are other factors that influence the work done, which is why they are path-Dependent (not Independent!) and cannot have arbitrary paths applied.

When discussing spring elasticity (stretching the spring), hookes law still applies to an extent. Describe the elastic region, plastic region, and breaking point when it comes to a force vs elongation graph

From 0-1 is the elastic region and in this region the force is directly proportional to our elongation From 1-2: After the elongation passes the limit of the elastic region, is the plastic region and in this region the hookes law doesn't 100 percent apply as there is not necessarily an proprtional increase in force as elongation increases. After 2 is something called the breaking point, and it pretty much means that is spring is stretched beyond this elongation point then it will break

What happens when the net work done on a moving object is 0, +, or -?

If positive, KE increases and object speeds up If negative, KE decrease and object slows down If 0, KE of that object remains the same and the object will maintain a constant speed

*CRB* Compare Elastic and Inelastic Collisions, focusing on how total momentum and total kinetic energy are affected.

In Elastic Collisions, both total momentum and total kinetic energy are conserved. In Inelastic Collisions, total momentum is still conserved, but total kinetic energy is not, since it is lost in the collision.

*CRB* What is the Conservation of Total Mechanical Energy Equation in its general form? How would you write it out using Kinetic and Potential Energy?

In its general form: Einitial = Efinal See image for other form.

What is the equation for kinetic energy in terms of velocity?

Kinetic Energy = (1/2)mv^2 m = mass v = velocity Energy or work needed to accelerate an object from being stationary to its current velocity F*d= (1/2)mv^2 ...because by definition, kinetic energy is the amount of work you need to put into an object to get it from rest to its current velocity ( how much work was put in to accelerate object to its current velocity)

*CRB* Compare Angular and Linear Momentum.

Linear Momentum is what has been already discussed in these cards, p=mv. Angular momentum is similar to torque, in the way that it relates to some reference point, and could be written L=lmv, with l being related to torque's lever arm.

What is the equation for mechanical advantage in terms of force in and out?

MA = Fout/Fin Fin = Force in Fout = Force out MA = Mechanical Advantage

What is the units of the spring constant?

N/m so it tells you how much newtons of force is needed to stretch or compress the spring 1 m

When is work negative vs positive?

Negative= when force is done in opposite direction of displacement. Work would be seen as -f*d and energy would be taken away from the object Positive= when force is done in same direction of displacement Zero= when force is done perpendicular to the displacement, or if object does not move. When the force is applied perpendicular to an object, it is neither taking away or giving energy to that object

*CRB* If the Non-Conservative Force of Friction is acting on an object, then how would you write out the Conservation of Mechanical Energy Equation?

Note: Any Non-Conservative force could be in the place of friction here.

When you put work into an object you put what into the object?

Once you overcome other forces such as friction, etc.. the work remaining is used to put energy into the object that can either be converted to potential or kinetic energy. A moving object will gain kinetic energy.

What is the equation for gravitational potential energy?

PE = mgh F*d= mgh m = mass g = acceleration due to gravity h = height at the object is at Work had to be put into the object to get it to that certain height.. but object now is not moving ( the work done on the object to get it to a certain height by definition was done under a constant velocity so there was no work applied to an object to increase its velocity so there is no kinetic energy) so its not under kinetic energy but it still has the potential to do work. Gravitational potential energy is equal to the amount of work if took to move object against the force of gravity to get it to that height. SO W= Fd and in this case the force is (m*g) and the distance is h, so W=PE=mgh

What is the equation for the potential energy of a spring?

PEs = 1/2 k x^2 PEs = Potential energy of a spring x = displacement k = spring constant

If a force is conservative, you can find what for that force?

Potential energy. That's why gravitational force and spring force have PEs. Example if you do work on a spring, you can get that work/energy back by letting the spring do back the work on you which turns that stored PE in spring into KE that it gave the object it worked on. But if you do work against the force of friction (non conservative), it will be hard to get that energy back, the energy is dissipated as formal energy

What is the equation for instantaneous power in terms of velocity?

Power = Force x Velocity This comes from work/time= (fdcos(thetha))/time and converting d/t to v If instantaneous power remains same through the whole motion then the avg power = instantaneous

What is the equation for power in terms of work?

Power is the rate at which someone or something does work Power = Work / Time

Find the work done in all directions of a trashcan that moves 10m with a force of 50 N applied and a frictional force of 30 N applied in the opposite direction. Use these values to find the speed of the trash can after it slid 10 m using the work energy principle

Recall that work = Fdcos(theta) 500 J of work is applied Friction applies -300 J of work (work done in opposite direction of movement) Overall there is a net work of 200 N or 200 J of energy is applied to object Both gravity and Normal force do 0 work because force is being applied perpendicular to the movement of the object. Note that we assumed that the object started off wit 0 KE

AK lecture example of gravity being a conservative force:

Since gravity is a conservative force, regardless of the path taken, the work done by gravity is the same. For ball 1.. force points in same direction as displacement so to find the work done by gravity or the amount of potential energy that was stored in the ball, the formula is Wgravity= mgh For gravity to be a conservative force, case 2 should also give us the same work. We must find the length of the incline plane and go from there.

*CRB* In the previous example, how much work was done by the Normal Force? How much work was done by Gravity? Explain.

There was no vertical displacement due to the Normal Force or Gravity. Because there was no displacement, the work done by these forces is 0!

Describe how the work formula would look like had the force not been directly applied in the direction of the displacement but at an angle?

There would be a multiple of a trig number and theta, because truly the only force that impacts the value of work is the force going in the direction of the displacement

*CRB* There are also Perfectly Inelastic Collisions. What makes these collisions unique?

These inelastic conditions have the objects stick together after the collision! So their velocity will be the same vector.

What is significant about Work having the same units as energy? N*m or Joules

This makes sense because work is telling you the amount of joules of energy given to or taken away from the system

*CRB* True or false? If there is an obtuse angle (more than 90 less than 180) between Force and Displacement, then the Work done by the force must be negative.

True. If there is an obtuse angle between Force and Displacement, then the Work done by the force must be negative. This is because the cosine of the angle is negative! Recall the unit circle where cosine of anything in quadrant 2 is negative

*CRB* True or false? The equation for Mechanical Advantage could also be written MA= (Resistance Force) / (Effort Force)

True. The equation for Mechanical Advantage could also be written MA= (Resistance Force) / (Effort Force)

*CRB* True or false? When you are dealing with Conservative Forces, you can treat the force and path of object as independent and assign any arbitrary path to the object. Why would or wouldn't this be true for Non-Conservative

True. When you are dealing with Conservative Forces, you can treat the force and path of object as independent and assign any arbitrary path to the object.

What is the equation for work in terms of force?

Work = Force x Distance Work is a process in which energy is transferred from one system to another, and energy is the ability to do work Note work is translation movement and torque is rotational movement

*CRB* There will also be some instances where the constant force and the displacement are not in the same direction. In this case, what is the equation for Work?

Work = Force x Distance traveled by object x cosθ Work tells you the amount of energy that the force F is giving to an object

What is the Net Work-Energy principle? (Work in terms of Kinetic energy) What is the net work done on an object:

Work = change in kinetic energy The amount of net work done on an object is equal too the final amount of energy (work that was put into the object) minus the amount of energy that object already had. So if does work on an object and gives object less energy than it already had, net work would be negative and object would be going in opposite direction of applied force.

What is the difference between a conservative force and a non-conservative force?

Work done on force only depends on initial and final position. A conservative force conserves mechanical energy (KE +PE), and because of this it doesn't matter what path it takes to calculate work, just the beginning and ending position. A non conservative force doesn't conserve mechanical energy, and the energy cannot be regained by reversing the process ( like how we could do with raising and dropping an object with gaining and regaining that potential energy) , because of this the path affects the work done.

What is the equation that relates the forces in and out when dealing with mechanical advantage in terms of distance of forces from a fulcrum?

f1 x d1 = f2 x d2 F1 = force 1 d1 = distance from force 1 to fulcrum f2 = force 2 d2 = distance from force 2 to fulcrum this pretty much shows force in must equal force out in this case. The actual distance that work is being done on is the distance of the red arrow but due to proportionality of the red and blue arrows for there distances from the fulcrum, then this formula can be used

*CRB* Unrelated to Moment, what is the equation for Momentum?

p = mv p = Momentum m = Mass v = Velocity

If work is done at a constant velocity on the trashcan to move it upwards 2m, find what the net work (Wnet) on that object is

recall that Work = Fdcos(theta) TO move the trashcan upwards the amount of work done is 78.4 J because the force is applied the same direction as object moves (cos (0)) The work done by gravity however is -78.4 J because the work is being done in the opposite direction of the movement of the object. This will give a Wnet or Change in KE of 0 which makes sense because something moving at a constant velocity should not have KE because it is not accelerating