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**Question-1**. A box of mass m is sliding along a horizontal surface.

**Part A** The box leaves position x=0 with speed v0. The box is slowed by a constant frictional force until it comes to rest at position x=x1.

Find the magnitude of the average frictional force that acts on the box. (Since you don’t know the coefficient of friction, don’t include it in your answer.)**Express the frictional force in terms of m, v0, x1**

**Part B** After the box comes to rest at position x1, a person starts pushing the box, giving it a speed v1.

When the box reaches position x2(wherex2 > x1), how much work has the person done on the box? Assume that the box reaches x2 after the person has accelerated it from rest to speedv1.**Express the work in terms of m, v0, x1, x2, v1**

**Question-2**

Understand that conservative forces can be removed from the work integral by incorporating them into a new form of energy called potential energy that must be added to the kinetic energy to get the total mechanical energy.

The first part of this problem contains short-answer questions that review the work-energy theorem. In the second part we introduce the concept of potential energy. But for now, please answer in terms of the work-energy theorem.(PART-A to PART E)

**Question-3.**

Part A Consider a uniform gravitational field (a fair approximation near the surface of a planet). Find U(yf)−U(y0)=−∫yfy0F⃗ g⋅ds⃗ , where F⃗ g=−mgj^ and ds⃗ =dyj^. Express your answer in terms of m, g, y0, and yf. U(yf)−U(y0) = SubmitHintsMy AnswersGive UpReview Part Part B Consider the force exerted by a spring that obeys Hooke’s law. Find U(xf)−U(x0)=−∫xfx0F⃗ s⋅ds⃗ , where F⃗ s=−kxi^,ds⃗ =dxi^, and the spring constant k is positive. Express your answer in terms of k, x0, and xf. U(xf)−U(x0) = SubmitHintsMy AnswersGive UpReview Part Part C Finally, consider the gravitational force generated by a spherically symmetrical massive object. The magnitude and direction of such a force are given by Newton’s law of gravity: F⃗ G=−Gm1m2r2r^, where ds⃗ =drr^; G, m1, and m2 are constants; and r>0. Find U(rf)−U(r0)=−∫rfr0F⃗ G⋅ds⃗ . Express your answer in terms of G, m1, m2, r0, and rf. U(rf)−U(r0) = SubmitHintsMy AnswersGive UpReview Part

**Question 4.**

Suppose our experimenter repeats his experiment on a planet more massive than Earth, where the acceleration due to gravity is *g*=30 m/s2. When he releases the ball from chin height without giving it a push, how will the ball’s behavior differ from its behavior on Earth? Ignore friction and air resistance. (Select all that apply.)

**Question 5.**

While a roofer is working on a roof that slants at 41.0 ∘ above the horizontal, he accidentally nudges his 95.0 N tool box, causing it to start sliding downward, starting from rest.

**Part A** If it starts 5.00 m from the lower edge of the roof, how fast will the toolbox be moving just as it reaches the edge of the roof if the kinetic friction force on it is 21.0 N?

**Question 6.**

A force parallel to the *x*-axis acts on a particle moving along the *x*-axis. This force produces a potential energy *U*(*x*) given by *U*(*x*)=*α* *x*4where *α*=1.16 J/m4 .

**Part A **What is the force when the particle is at position *x* = -0.660 m ?

**Question 7**

As you are trying to move a heavy box of mass *m*, you realize that it is too heavy for you to lift by yourself. There is no one around to help, so you attach an ideal pulley to the box and a massless rope to the ceiling, which you wrap around the pulley. You pull up on the rope to lift the box.

()

Use *g* for the magnitude of the acceleration due to gravity and neglect friction forces.

**Part A **Once you have pulled hard enough to start the box moving upward, what is the magnitude *F* of the upward force you must apply to the rope to start raising the box with constant velocity?

**Express the magnitude of the force in terms of** *m***, the mass of the box.**

**Part B**Consider lifting a box of mass to a height using two different methods: lifting the box directly or liftingthe box using a pulley (as in the previous part).

What is , the ratio of the work done lifting the box directly to the work done lifting the box with a pulley?

Express the ratio numerically.

**Question 8**

To be able to interpret potential energy diagrams and predict the corresponding motion of a particle.

Potential energy diagrams for a particle are useful in predicting the motion of that particle. These diagrams allow one to determine the direction of the force acting on the particle at any point, the points of stable and unstable equilibrium, the particle’s kinetic energy, etc.

Consider the potential energy diagram shown. () The curve represents the value of potential energy *U* as a function of the particle’s coordinate *x*. The horizontal line above the curve represents the constant value of the total energy of the particle *E*. The total energy *E* is the sum of kinetic ( *K*) and ….

part A

The force acting on the particle at point A is __________.

part B

The force acting on the particle at point Cis __________.

par C

The force acting on the particle at point Bis __________.

part D

The acceleration of the particle at point Bis __________.

part E

If the particle is located slightly to theleft of point B, its acceleration is __________.

part F

If the particle is located slightly to theright of point B, its acceleration is __________.

part G

Name all labeled points on the graphcorresponding to *unstable* equilibrium.

List your choices alphabetically,with no commas or spaces; for instance, if you choose points B, D,and E, type your answer as BDE.

part H

Name all labeled points on the graphcorresponding to *stable* equilibrium.

List your choices alphabetically,with no commas or spaces; for instance, if you choose points B, D,and E, type your answer as BDE

part I

Name all labeled points on the graph wherethe acceleration of the particle is zero.

List your choices alphabetically,with no commas or spaces; for instance, if you choose points B, D,and E, type your answer as BDE

part J

Name all labeled points such that when aparticle is released from rest there, it would accelerate to theleft.

List your choices alphabetically,with no commas or spaces; for instance, if you choose points B, D,and E, type your answer as BDE.

part K

Consider points A, E, and G. Of these threepoints, which one corresponds to the greatest magnitude ofacceleration of the particle?

part L

What point on the graph corresponds to themaximum kinetic energy of the moving particle?

part M

At what point on the graph does the particlehave the lowest speed?

**Question 9**

A roller coaster car may be approximated by a block of mass (m) . The car, which starts from rest, is released at a height (h) above the ground and slides along a frictionless track. The car encounters a loop of radius (R), as shown. Assume that the initial height (h) is great enough so that the car never loses contact with the track.

Find an expression for the kinetic energy of the car at the top of the loop.

Express the kinetic energy in terms of m,g,h and R

Find the minimum initial height (h) at which the car can be released that still allows the car to stay in contact with the track at the top of the loop.

Express the minimum height in terms of R

**Question 10**

If a fish is attached to a vertical spring and slowly lowered to its equilibrium position, it is found to stretch the spring by an amount *d*.

If the same fish is attached to the end of the unstretched spring and then allowed to fall from rest, through what maximum distance does it stretch the spring? (*Hint:* Calculate the force constant of the spring in terms of the distance *d* and the mass *m* of the fish.)

**Express your answer in terms of** *d***.**

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