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(1) A sailboat is sailing at a bearing of 35° North of East, while the wind is exerting a force of 750 Newtons to the North on the sails. How much work does the wind do while the sailboat travels 4 kilometers? Hint: draw the vectors and note that is the angle between the vectors when you put their tails together.

(1) A sailboat is sailing at a bearing of 35° North of East, while the wind is exerting a force of 750 Newtons to the North on the sails. How much work does the wind do while the sailboat travels 4 kilometers? Hint: draw the vectors and note that is the angle between the vectors when you put their tails together.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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(3) The power is the rate of work or rate of energy transfer. The SI units for power are joule/second also known as a watt. The power provided by a force Facting on a object moving at velocity is given by 3 d ar dW P= = at dt =(7·47)=77= 7.7 A force F = (3î - 2j)N acts on a particle traveling with velocity = (4 1 + 6j) m/s. a. What is the power generated by this force? b. Another way to calculate the scalar product is to use the angle between the vectors so P = Based on your result for part (a), what was the angle between F and ? F|||| cos . (4) The scalar product is also useful for applications other than finding the work done by a force. The scalar product makes some problems that would otherwise be difficult relatively easy. For example, the scalar product can be used to find the angle between two three dimensional vectors. Example: Suppose that the data from X-ray diffraction of a crystal indicates that the positions of the atoms in a molecule are as follows: The position of the carbon atom is = 0 + 0 + 0k, the position of a neighboring oxygen atom is Ō = 21 +2j+ , and a hydrogen atom is located at = -1 +2j-k. The oxygen atom and the hydrogen atoms are both bonded to the carbon atom. Distances are in nanometers (nm). a. What is the length ||0|| of the C-O bond? b. What is the length of the C-H bond? c. What is the angle 8 between the C-O and C-H bonds? You can solve this problem by taking the two ways to find the scalar product and setting them equal to each other, i.e., use the fact that -= OH cos 8 = 0₂H₂ + O₂H₂ + O₂H₂.
Transcribed Image Text:(3) The power is the rate of work or rate of energy transfer. The SI units for power are joule/second also known as a watt. The power provided by a force Facting on a object moving at velocity is given by 3 d ar dW P= = at dt =(7·47)=77= 7.7 A force F = (3î - 2j)N acts on a particle traveling with velocity = (4 1 + 6j) m/s. a. What is the power generated by this force? b. Another way to calculate the scalar product is to use the angle between the vectors so P = Based on your result for part (a), what was the angle between F and ? F|||| cos . (4) The scalar product is also useful for applications other than finding the work done by a force. The scalar product makes some problems that would otherwise be difficult relatively easy. For example, the scalar product can be used to find the angle between two three dimensional vectors. Example: Suppose that the data from X-ray diffraction of a crystal indicates that the positions of the atoms in a molecule are as follows: The position of the carbon atom is = 0 + 0 + 0k, the position of a neighboring oxygen atom is Ō = 21 +2j+ , and a hydrogen atom is located at = -1 +2j-k. The oxygen atom and the hydrogen atoms are both bonded to the carbon atom. Distances are in nanometers (nm). a. What is the length ||0|| of the C-O bond? b. What is the length of the C-H bond? c. What is the angle 8 between the C-O and C-H bonds? You can solve this problem by taking the two ways to find the scalar product and setting them equal to each other, i.e., use the fact that -= OH cos 8 = 0₂H₂ + O₂H₂ + O₂H₂.
Activity 3.1 - Scalar Product Worksheet Objective - (A) Learn how to calculate the scalar product with (i) vectors in magnitude-direction form, and (ii) vectors in component unit-vector form. (B) Learn to use the scalar product as a tool in various applications. The scalar product of two vectors is also commonly called the "dot" product or the "inner product". By definition, the scalar product of two vectors A and B is the product of (1) the component of vector that is parallel to the vector B and (2) the magnitude of the vector B. Hence, the scalar product of the two vectors is a scalar quantity. This leads to the first way to calculate the scalar product. A B The angle between the vectors A and B is e. What is All, the component of A parallel to ? Write answer in terms of the magnitude A = A and angle 8. By definition A - B = AB, therefore the scalar product can also be written AB = AB cos 8. Work by constant force: The work done by a constant force is equal to the scalar product of the force and the displacement of the object it is acting on, i.e., W=F-AF. Notice that work is a scalar. (1) A sailboat is sailing at a bearing of 35° North of East, while the wind is exerting a force of 750 Newtons to the North on the sails. How much work does the wind do while the sailboat travels 4 kilometers? Hint: draw the vectors and note that is the angle between the vectors when you put their tails together. For three dimensional vectors Ā= A¸î + Ayĵ+ A₂k and B = B₂₁ + Bj + B₂k, the scalar product written in terms of components will be A - B = A₂B₂+AyBy + A₂B₂. This gives us two ways to calculate the scalar product (i) using the magnitude of the vectors and the angle between the vectors and (ii) using the components of the vectors. We choose whichever method is easier for the particular problem. (2) The combined force of lift and thrust on a model airplane is a constant = (4î +3j + 6k) N. a. How much work is done by the force F if the displacement of the airplane is: AF = (6î - 5j - 3k) m? b. Will this force cause the airplane to speed up, slow down or maintain a constant speed?
Transcribed Image Text:Activity 3.1 - Scalar Product Worksheet Objective - (A) Learn how to calculate the scalar product with (i) vectors in magnitude-direction form, and (ii) vectors in component unit-vector form. (B) Learn to use the scalar product as a tool in various applications. The scalar product of two vectors is also commonly called the "dot" product or the "inner product". By definition, the scalar product of two vectors A and B is the product of (1) the component of vector that is parallel to the vector B and (2) the magnitude of the vector B. Hence, the scalar product of the two vectors is a scalar quantity. This leads to the first way to calculate the scalar product. A B The angle between the vectors A and B is e. What is All, the component of A parallel to ? Write answer in terms of the magnitude A = A and angle 8. By definition A - B = AB, therefore the scalar product can also be written AB = AB cos 8. Work by constant force: The work done by a constant force is equal to the scalar product of the force and the displacement of the object it is acting on, i.e., W=F-AF. Notice that work is a scalar. (1) A sailboat is sailing at a bearing of 35° North of East, while the wind is exerting a force of 750 Newtons to the North on the sails. How much work does the wind do while the sailboat travels 4 kilometers? Hint: draw the vectors and note that is the angle between the vectors when you put their tails together. For three dimensional vectors Ā= A¸î + Ayĵ+ A₂k and B = B₂₁ + Bj + B₂k, the scalar product written in terms of components will be A - B = A₂B₂+AyBy + A₂B₂. This gives us two ways to calculate the scalar product (i) using the magnitude of the vectors and the angle between the vectors and (ii) using the components of the vectors. We choose whichever method is easier for the particular problem. (2) The combined force of lift and thrust on a model airplane is a constant = (4î +3j + 6k) N. a. How much work is done by the force F if the displacement of the airplane is: AF = (6î - 5j - 3k) m? b. Will this force cause the airplane to speed up, slow down or maintain a constant speed?
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