Ch 07 HW Part 2 - Chapter 7 Physics Homework for Mastering PDF

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Chapter 7 Physics Homework for Mastering
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Ch 07 HW Part 2 Due: 3:00pm on Wednesday, October 30, 2019 You will receive no credit for items you complete after the assignment is due. Grading Policy

Problem 7.38 Froghoppers may be the insect jumping champs. These 6-long bugs can spring 70 into the air, about the same distance as the flea. But the froghopper is 60 times more massive than a 12flea. The froghopper pushes off for about 4 .

Part A Determine the average force that the froghopper exerts on the surface. Express your answer with the appropriate units. ANSWER: = 1.2

All attempts used; correct answer displayed

Part B Compare this to the gravitational force that Earth exerts on the bug. ANSWER:

This is about 176 times less than the weight of the froghopper. This is about 176 times the weight of the froghopper. This is equal to the weight of the froghopper.

Correct

Problem 7.44 Your friend provides a solution to the following problem. Evaluate her solution. The problem: Jim (mass 50 ) steps off a ledge that is 2.0 above a platform that sits on top of a relaxed spring of force constant 8000 . How far will the spring compress while stopping Jim? Your friend's solution:

Part A Is this answer correct? ANSWER:

yes no

Correct

Part B Determine the correct spring compression. Express your answer with the appropriate units. ANSWER: = 0.56

All attempts used; correct answer displayed

Problem 7.52 Somebody tells you that shows a displacement-versus-time graph of two carts on a linear track before and after a collision, when friction and air drag are negligible. The mass of cart is 2.0 and mass of cart is 1.0 .

Part A Is the graph in agreement with the laws of physics? ANSWER:

Yes No

Correct

Part B Determine the kinetic energy of the system before the collision. Express your answer with the appropriate units.

ANSWER: = 0.13

Correct

Part C Determine the kinetic energy of the system after the collision. Express your answer with the appropriate units. ANSWER: = 9.5×10−2

All attempts used; correct answer displayed

Part D Is the collision elastic, inelastic, or totally inelastic? ANSWER:

elastic inelastic totally inelastic

Correct

Problem 7.68

Part A If the Sun were to become a black hole, how much would it increase the gravitational potential energy of the Sun-Earth system? ANSWER: The gravitational potential energy would increase by 4 times. The gravitational potential energy would become zero. The gravitational potential energy would not change. The gravitational potential energy would double.

Correct

Problem 7.72

A spherical street lamp accidentally explodes. Three equal pieces directions, as shown in .

,

, and

fly off the lamp holder with equal speeds but in different

Part A Compare the speeds with which each piece hits the ground. Neglect the size of the lamp and air resistance. ANSWER:

Correct

Part B Compare qualitatively the times needed for each piece to reach the ground. ANSWER:

Correct

Conceptual Question 7.20

Part A What will happen to Earth if our Sun becomes a black hole (with the same mass)? Check all that apply.

ANSWER: Earth will fall on Sun. The Earth's orbit will change. Earth will be freezing in the absence of a heat source. Nothing will happen.

Correct

Direct Measurement Video: Conservation Laws in a Real Collision - Part 1 We will use this slow-motion video (Text description of Direct Measurement Video) to explore conservation of linear momentum and conservation of kinetic energy to collisions.

Specifically, we will: 1. Define a specific system that is most useful for our analysis. 2. Take measurements to calculate the change in linear momentum of a system as a result of a collision. 3. Take measurements to calculate the change in kinetic energy of a system as a result of a collision. The event we will analyze is a heavy iron weight-lifting disk coming into contact with a rolling low-friction cart after it is dropped vertically onto the cart. After bouncing and sliding across the cart’s surface, the disk comes to rest with respect to the cart’s surface, and the disk and cart move together. To simplify and clarify this analysis, we’ll consider the horizontal motion only. Would you predict that linear momentum and/or kinetic energy are conserved during this collision? Knowing when and how to apply the conservation laws for these two quantities is an important part of analyzing interactions between objects. Let’s do some analysis of this interesting situation to see whether and why linear momentum and kinetic energy are conserved or not conserved.

Part A - Define a specific system that is most useful for our analysis. Let’s start by defining a useful system to analyze this interaction. After watching the video, what system do you think would be the most useful when analyzing the collision between the disk and the cart? Select the best answer from the choices provided.

Hint 1. Defining a system The advantage of analyzing a system rather than an individual object is that we can gain important insights about how quantities like the energy and momentum of a group of objects change during interactions such as collisions.

ANSWER:

the cart only the disk, cart, floor, and the earth the disk and cart the disk only the disk, cart, and floor

Correct There are many possible choices for a system in this situation. For example, in addition to the cart and the iron disk, we could include the earth. But in this case, since we are concerned with the horizontal motion only and the earth applies only vertical forces, we won’t include the earth. We could also include the floor in our system, but we can’t make any measurements of the motion of the floor or the forces exerted by the floor. Let’s use only the cart and the disk as our agreed-upon system. This means that we’ll have to consider forces due to the earth or the floor, such as the normal force and friction exerted by the floor on the cart wheels, to be external to our system. Now that we have defined a system useful for our analysis, let’s make measurements from the video that we can use to determine the change in momentum and kinetic energy of the system before and after the collision. We’ll start by determining these quantities before the collision.

Analyzing the motion before the collision We’ll begin by making measurements from the video that we can use to determine the average velocity of the cart just before the disk lands on it. First watch the entire video. A white ruler is overlaid on the video as the cart approaches the disk. Notice that the video’s frame counter reads when the right edge of the blue dot on the right side of the cart is very near the mark on the white ruler. The video player has features that make it easy to navigate the video frame-by-frame: Use the spacebar to pause or resume playing the video When the video is paused, the right arrow keys on your keyboard will advance the video one frame. The left arrow key moves the video back one frame.

Part B Determine the average velocity of the cart during the time interval from frame convention that right is positive and left is negative.

to frame

. For all vector quantities, we’ll use the

Express your answer using SI units to three significant figures.

Hint 1. Measuring velocity from the video Average velocity is

. To determine velocity from the video, you’ll need to measure positions and time to determine

displacement and a time interval from the video. You can use the white rulers overlaid near the beginning and end of the video to measure position and the frame counter to measure a times. The video is recorded in slow motion. There is of a second of elapsed time between each frame. Hint 2. Measuring the time interval Note that a “frame” is a still image in the video. The frame counter labels the frames consecutively to allow us to count the frames during a specific segment of the video. Because the video is made up of frames per second, the term “frame” can also be used to describe the time interval between consecutive frames (images). A frame in this sense is therefore a unit of time. A time interval is . To determine the time interval required for this motion, you need to note the final and the initial times. These can be found in frames from the video’s frame counter. Since there is of a second of elapsed time between each frame, you can convert the number of frames elapsed to a time interval in seconds using the conversion factor:

. For example, if the time interval begins when the frame counter reads

and ends when it reads

, the

we’d say:

Hint 3. Measuring the cart’s displacement You can use the white ruler overlaid on the video along with the blue dot on the cart to measure the position of the cart at different times. Be consistent with your point of reference when making this measurement. Looking at the image below, we see that the right edge of the blue dot on the right side of the cart is very near the mark on the ruler at frame . Use the right edge of the blue dot again for subsequent position measurements. Displacement is:

where

ANSWER: 1.80

and

are the final and initial positions of the cart

Correct Let's make sure we've got this right before moving ahead. Looking at the image below, we see that the right edge of the blue dot on the right side of the cart is very near the mark on the ruler at frame .

The image below shows that at frame

, the right edge of that same dot is near the

mark on the ruler.

This gives a displacement of: (to the right, so positive) Our time interval for this motion is:

Finally, this gives a velocity of: (again, to the right and therefore positive) As for all measurements, there is some uncertainty about the exact value of this measurement. We’ll use this value for subsequent calculations.

Now that we know the velocity of the cart before the collision is of the disk-cart system before the collision. The mass of the cart is

, let’s use this to determine the linear momentum and kinetic energy . The mass of the disk is .

Part C Using this mass information and the velocity calculated, determine the horizontal component of momentum of the disk-cart system during this interval. Express your answer using appropriate units to three significant figures.

Hint 1. Calculating linear momentum The relationship we use to calculate the linear momentum of an object is object’s linear velocity.

, where

is the object’s mass and

is the

Hint 2. Multi-object systems Our system comprises the disk and the cart, so we need to add the horizontal components of the momenta of each to find the total system momentum. We’ve calculated the horizontal component of the velocity of the cart previously, so we can use that to determine the cart’s linear momentum. What is the horizontal component of the velocity of the disk before the collision? ANSWER: 0

Hint 3. Horizontal and vertical components of the momentum Our goal is to analyze the horizontal motion of the system during this collision. Although the disk does attain some momentum in the vertical direction as it falls, this motion does not affect the horizontal motion of the system.

ANSWER:

42.3

Correct The disk has no initial horizontal momentum so the system’s momentum is simply the cart’s momentum, calculated by multiplying its mass by its velocity. Again, this vector quantity is directed to the right, so we express it as a positive value.

Part D Next, we’ll continue by examining the kinetic energy of the disk-cart system before the collision. Determine the kinetic energy of the system during this interval. As a reminder, because we are analyzing the horizontal motion of the cart, we’ll measure only the horizontal velocities of the cart and disk. Express your answer using appropriate units to three significant figures.

Hint 1. Calculating kinetic energy The kinetic energy of an object of mass

and velocity

is

.

Hint 2. Multi-object systems Our system comprises the disk and the cart, so we need to add the kinetic energy of each to find the system's kinetic energy. We’ve calculated the velocity of the cart previously, so we can use that to determine the cart’s kinetic energy. What is the kinetic energy of the disk before the collision? Hint 3. Horizontal and vertical motion Our goal is to analyze the horizontal motion of the system during this collision. Although the disk does gain some kinetic energy as it falls, this motion does not affect the horizontal motion of the system.

ANSWER: 38.1

Correct The disk has negligible initial kinetic energy so the system’s kinetic energy is simply the cart’s kinetic energy, calculated by multiplying its mass by its velocity squared, all divided by two.

Analyzing the motion after the collision

We’ve completed our measurements and calculations for the velocity, linear momentum, and kinetic energy of the cart-disk system before the collision. Let’s repeat this process for the cart and disk as they move together after the collision. We’ll be able to make important observations by comparing before and after values for these quantities.

Part E Determine the average velocity of the cart and disk as they move together during a 24-frame interval after the collision. Use the blue dot on the left end of the cart as the point of reference for your measurement. Express your answer using SI units to three significant figures. ANSWER: 1.18

Correct As with the velocity before the collision, it is important that we agree on a value for the velocity of the cart and disk after the collision that we’ll use for the calculations to follow. Looking at the image below, we see that the right edge of the blue dot on the left side of the cart starts very near the line on the ruler when the frame counter shows .

The image below shows the right edge of the blue dot very near .

, or

, when the frame counter shows

This gives a displacement of:

Our time interval for this motion is:

This gives a velocity of:

As for all measurements, there is some uncertainty about the exact value of this measurement. We’ll use this value for subsequent calculations.

Part F Let’s use this velocity of to compare the total momentum and kinetic energy of the cart-disk system before and after the collision. First, let’s start with the momentum. Calculate the percent change in momentum of the cart-disk system resulting from the collision. Recall the mass of the disk is and the mass of the cart is . Express your answer as a positive or negative percentage to three significant figures, to indicate either an increase or decrease in momentum.

Hint 1. Multi-object systems Our system comprises the disk and the cart, so we need to add the horizontal components of momentum of each to find the total system momentum. We’ve calculated the velocities of the cart and disk previously, so we can use these to determine the momentum of the system. What is the momentum of the cart-disk system after the collision? Express your answer using appropriate units to three significant figures. ANSWER: 41.2

Hint 2. Calculating percent change The relationship for

.

ANSWER: -2.60

Correct The momentum of the system in this case decreases by around 3%. Next, we’ll repeat this process to compare the kinetic energy of the cart-disk system before and after the collision.

Part G Now that we’ve seen that the change in the momentum of the cart-disk system is a decrease of just a few percent, let’s compare the kinetic energy before and after the collision as well. Calculate the percent change in kinetic energy of the cart-disk system resulting from the collision. Express your answer as a positive or negative percentage to three significant figures, to indicate either an increase or decrease in kinetic energy.

Hint 1. Vertical and horizontal motion of the disk We can ignore the kinetic energy and momentum of the disk due to the vertical motion as it falls onto the cart. Firstly, the disk does not fall very far before landing on the cart. Secondly, and more importantly, the disk has a zero vertical component of velocity when it is released. Although it does have a vertical component of velocity when it lands, the normal force of the floor on the cart causes the vertical velocity to return to zero without affecting the horizontal motion of the disk or cart. Hint 2. Multi-object systems Our system comprises the disk and the cart, so we need to add the kinetic energy of each to find the total kinetic energy of the system. What is the kinetic energy of the cart-disk system after the collision? Express your answer using appropriate units to three significant figures. ANSWER:

24.3

Hint 3. Calculating percent change The relationship is \large{{\rm {Percentage\;change = 100\; \times \frac{(final\;value - initial\;value)} {initial\;value}\; \%}}

ANSWER: -36.2

Correct The kinetic energy of the system in this case decreases by around 36%. Let’s look at what we’ve done. We used the video to measure the velocity of the objects in the system we defined, before and after the collision. We’ve calculated the amount of momentum and kinetic energy for the system, before and after the collision. We’ve calculated the percent change of the momentum and kinetic energy of the system as a result of the collision. Notice that the percent change in kinetic energy value is much different than the change in momentum.

Direct Measurement Video: Conservation Laws in a Real Collision - Part 2 We will use this slow-motion video (Text description of Direct Measurement Video) to explore how and when we can apply conservation of linear momentum and conservation of kinetic energy to collisions. Our objective is to analyze a concrete example of how forces within a system affect conservation of kinetic energy and linear momentum during a collision.

Specifically, we will: 1. Define a specific system that is most useful for our analysis. 2. Analyze the types of forces acting on a system and internally within a system. 3. Define the conditions that must be met so that linear momentum and kinetic energy do not change as a result of a collision. 4. Apply mechanical energy and momentum conservation conditions to a collision and use them to explain the results of measurements. The event we will analyze is a heavy iron weight-lifting disk coming into contact with a rolling low-friction cart after it is dropped vertically onto the cart. After bouncing and sliding across the cart’s surface, the disk comes to rest with respect to the cart’s surface, and the disk and cart move together. To simplify and clarify this analysis, we’ll consider the horizontal motion only. Would you predict that linear momentum and/or kinetic energy are conserved during this collision? Knowing when and how to apply the conservation laws for these two quantities is an important part of analyzing interactions between objects. Let’s do some analysis of this interesting situation to see whether and why linear momentum and kinetic energy are conserved or not conserved.

Part A - Define a specific system that is most useful for our analysis. Let’s start by defining a useful system to analyze this interaction...