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Class PH1110A2007
Assignment 6
Due at 5:00pm on Monday, September 10, 2007
View Grading Details
Exercise 4.26
An athlete throws a ball of mass Part A Draw a free-body diagram of this ball while it is free of the athlete's hand and moving upward; Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: directly upward, and it feels no appreciable air resistance.
View Part B Draw a free-body diagram of this ball while it is free of the athlete's hand and at its highest point; Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER:
View Part C Draw a free-body diagram of this ball while it is free of the athlete's hand and moving downward. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER:
View Part D
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Repeat part A if the athlete throws the ball at a 60 angle above the horizontal instead of directly upward. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER:
View Part E Repeat part B if the athlete throws the ball at a 60 angle above the horizontal instead of directly upward. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER:
View Part F Repeat part C if the athlete throws the ball at a 60 angle above the horizontal instead of directly upward. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER:
View
Exercise 4.24
The upward normal force exerted by the floor is 620 on an elevator passenger who weighs 650 . Part A What is the magnitude of the acceleration? ANSWER: Part B What is the direction of the acceleration? = 0.452
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ANSWER:
upward downward
Pushing a Chair along the Floor
A chair of weight 115 lies atop a horizontal floor; the floor is not frictionless. You push on the chair with a force of = 41.0 directed at an angle of 39.0 below the horizontal and the chair slides along the floor. Part A Using Newton's laws, calculate , the magnitude of the normal force that the floor exerts on the chair. Hint A.1 How to approach the problem To solve this problem you need to focus on the forces that have a vertical component. In fact, because the direction of motion is horizontal (the chair slides along the floor), both the acceleration of the chair and the net force acting on the chair are purely horizontal. This can be true only if the net vertical force acting on the chair is zero, that is, if the normal force exerted on the chair by the floor balances all the other vertical forces acting on the chair. Also, in any problem involving forces you should always draw a free-body diagram that shows all the forces acting on the system. To do that, choose a coordinate system and identify all the forces acting on the chair. Part A.2 Choosing the correct free-body diagram Let be the friction force, be the force you exert on the chair, and be the weight of the chair. If you choose the x axis to be parallel to the floor, which free-body diagram would correctly represent the situation described in the introduction of this problem?
Hint A.2.a Direction of the friction force Hint not displayed
b c d
Part A.3 Find the vertical net force Let the positive y axis point upward, and let be the magnitude of the normal force exerted on the chair by the floor, be the magnitude of the weight of the chair, and be the magnitude of the y component of the force you exert on the chair when you push it. What is the vertical net force, that is, the y component of the net force, acting on the chair? Hint A.3.a How to approach the problem Hint not displayed ANSWER:
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Recall that the vertical net force is zero because there is no vertical acceleration. To find the normal force exerted on the chair by the floor, find the vertical component of the force you exert on the chair and solve for . Part A.4 Find the vertical component of the force that you exert on the chair Find , the magnitude of the vertical component of the force that you exert on the chair when you push it.
Hint A.4.a Components of a vector Consider a vector respectively, and where is the magnitude of the vector. that forms an angle with the positive x axis. The x and the y components of are,
Express your answer in newtons. ANSWER: = 25.8
Express your answer in newtons. ANSWER: = 141
Understanding Newton's Laws
Part A An object cannot remain at rest unless which of the following holds? Hint A.1 How to approach the problem Hint not displayed Hint A.2 Newton's 1st law: a body at rest Hint not displayed ANSWER: The net force acting on it is zero. The net force acting on it is constant and nonzero. There are no forces at all acting on it. There is only one force acting on it.
If there is a net force acting on a body, regardless of whether it is a constant force, the body accelerates. If the body is at rest and the net force acting on it is zero, then it will remain at rest. The net force could be zero either because there are no forces acting on the body at all or because several forces are acting on the body but they all cancel out. Part B
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If a block is moving to the left at a constant velocity, what can one conclude? Hint B.1 How to approach the problem Hint not displayed Hint B.2 Newton's 1st law: a body in motion Hint not displayed ANSWER: There is exactly one force applied to the block. The net force applied to the block is directed to the left. The net force applied to the block is zero. There must be no forces at all applied to the block.
If there is a net force acting on a body, regardless of whether the body is already moving, the body accelerates. If a body is moving with constant velocity, then it is not accelerating and the net force acting on it is zero. The net force could be zero either because there are no forces acting on the body at all or because several forces are acting on the body but they all cancel out. Part C A block of mass is acted upon by two forces: say about the block's motion? Hint C.1 How to approach the problem Hint not displayed Hint C.2 Newton's 2nd law Hint not displayed Hint C.3 Relating acceleration to velocity Hint not displayed ANSWER: It must be moving to the left. It must be moving to the right. It must be at rest. It could be moving to the left, moving to the right, or be instantaneously at rest. (directed to the left) and (directed to the right). What can you
The acceleration of an tells object you nothing about its velocity--the direction and speed at which it is moving. In this case, the net force on (and therefore the acceleration of) the block is to the right, but the block could be moving left, right, or in any other direction. Part D A massive block is being pulled along a horizontal frictionless surface by a constant horizontal force. The block must be __________. Hint D.1 How to approach the problem Hint not displayed Hint D.2 Newton's 2nd law Hint not displayed ANSWER: continuously changing direction
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moving at constant velocity moving with a constant nonzero acceleration moving with continuously increasing acceleration Since there is a net force acting, the body does not move at a constant velocity, but it accelerates instead. However, the force acting on the body is constant. Hence, according to Newton's 2nd law of motion, the acceleration of the body is also constant. Part E Two forces, of magnitude and , are applied to an object. The relative direction of the forces is unknown. The net force acting on the object __________. A. cannot be equal to B. cannot be equal to C. cannot be directed the same way as the force of D. must be greater than Hint E.1 How to approach the problem By definition, the net force is the vector sum of all forces acting on the object. To find the magnitude of the net force you need to add the components of the two forces acting. Try adding the two forces graphically (by connecting the head of one force to the tail of the other). The directions of the two forces are arbitrary, but by trying different possibilities you should be able to determine the maximum and minimum net forces that could act on the object. Part E.2 Find the net force when the two forces act on the object in opposite directions Find the magnitude of the net force if both the forces acting on the object are horizontal and the 10-N force is directed to the right, while the 4-N force is directed to the left. Hint E.2.a Vector addition Hint not displayed Express your answer in newtons. ANSWER: 6.0
Is there any other orientation of the two forces that would lead to a net force with a smaller magnitude than what you just calculated? Part E.3 Find the direction of the net force when the two forces act in opposite directions If both the forces acting on the object are horizontal and the 10-N force is directed to the right, while the 4-N force is directed to the left, the net force is horizontal and directed __________. ANSWER: in the same direction as the 10-N force in the opposite direction to the 10-N force
Enter the letters of all the correct answers in alphabetical order. Do not use commas. For example, if you think only the last option is correct, enter D .
Free-Body Diagrams and Newton's Laws
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When solving problems involving forces and Newton's laws, the following summary of things to do will start your mind thinking about getting involved in the problem at hand. Problem Solving: Free-Body Diagrams and Newton's Laws 1. Draw a sketch of the situation. 2. Consider only one object (at a time), and draw a free-body diagram for that body, showing all the forces acting on that body. Do not show any forces that the body exerts on other bodies. If several bodies are involved, draw a free-body diagram for each body separately, showing all the forces acting on that body. 3. Newton's second law involves vectors, and it is usually important to resolve vectors into components. Choose an x and y axis in a way that simplifies the calculation. 4. For each body, Newton's second law can be applied to the x and y components separately. That is the x component of the net force on that body will be related to the x component of that body's acceleration: , and similarly for the y direction. 5. Solve the equation or equations for the unknown(s). Apply these steps Use the steps outlined above to find the magnitude of the acceleration of a chair and the magnitude of the normal force acting on the chair: Yusef pushes a chair of mass = 55.0 across a carpeted floor with a force of magnitude = 154 directed at = 35.0 the carpet and the floor is = 107 . below the horiztonal . The magnitude of the frictional force between
Part A Identify and sketch all the external forces acting on the chair. Because the chair can be represented as a point particle of mass , draw the forces with their tails centered on the black dot in the middle of the chair. Be certain to draw your forces so that they have the correct orientation. ANSWER:
View Part B Which set of coordinate axes is the most convenient to use in this problem? Part B.1 Determine the direction of the acceleration Part not displayed
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ANSWER:
Now that you have selected a coordinate system, you should resolve the forces into x and y components so that you can apply Newton's second law to each coordinate direction independently. Part C Use the component form of Newton's second law to write an expression for the x component of the net force, Part C.1 Find the x component of the pushing force What is , the x component of the force exerted by Yusef? .
Hint C.1.a Right angle triangle trigonometry Hint not displayed Express your answer in terms of the variables ANSWER: = , , , , and . and .
Express your answer in terms of some or all of the variables: ANSWER: Part D =
Use the component form of Newton's second law to write an expression for the y component of the net force, Part D.1 Find the y component of the pushing force What is , the y component of the force exerted by Yusef?
.
Hint D.1.a Right angle triangle trigonometry Hint not displayed Express your answer in terms of the variables ANSWER: = , , , , and . and .
Express your answer in terms of some or all of the variables: ANSWER: =
You have created two equations that describe the motion of the chair:
and
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Solve these equations to find Part E
and .
What is the magnitude of the acceleration of the chair? What is the magnitude of the normal force chair? Part E.1 Find the component of the acceleration in the y direction The chair only moves in the postive x direction. What is ? Express your answer in meters per second squared. ANSWER: = 0 and :
acting on the
The magnitude of the acceleration, is found from
Because
,
Part E.2 Find the weight of the chair What is the weight of the chair? Hint E.2.a An equation for weight Recall that the weight of an object of mass is given by , where is the acceleration due to gravity.
Express the weight in newtons to three significant figures. ANSWER: = 539 .
Now you can use the equation you obtained by summing the forces in the y direction to find
Express your answers, separated by a comma, in meters per second squared and newtons to three significant figures. ANSWER: , = 0.355,627 ,
A free-body diagram is a useful way to begin all problems involving forces. This drawing will help you to easily identify the most appropriate coordinate axes and to resolve any 2 dimensional vectors into components. Then you can apply Newton's second law to each coordinate direction to set up equations which will allow you to solve for any unknown quantities.
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