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Math Calculus EBK MULTIVARIABLE CALCULUS

If a particle with mass m moves with position vector r ( t ), then its angular momentum is defined as L ( t ) = m r ( t ) × v ( t ) and its torque as τ ( t ) = m r ( t ) × a ( t ). Show that L ′( t ) = τ ( t ). Deduce that if τ ( t ) = 0 for all t , then L ( t ) is constant. (This is the law of conservation of angular momentum .)

If a particle with mass m moves with position vector r ( t ), then its angular momentum is defined as L ( t ) = m r ( t ) × v ( t ) and its torque as τ ( t ) = m r ( t ) × a ( t ). Show that L ′( t ) = τ ( t ). Deduce that if τ ( t ) = 0 for all t , then L ( t ) is constant. (This is the law of conservation of angular momentum .)

Solution Summary: The author explains the formula for differentiation with respect to t on both sides of equation (1).

EBK MULTIVARIABLE CALCULUS

EBK MULTIVARIABLE CALCULUS

8th Edition

ISBN: 9780100807884

Author: Stewart

Publisher: YUZU

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Chapter 13.4, Problem 44E

If a particle with mass m moves with position vector r(t), then its angular momentum is defined as L(t) = mr(t) × v(t) and its torque as τ(t) = mr(t) × a(t). Show that L′(t) = τ(t). Deduce that if τ(t) = 0 for all t, then L(t) is constant. (This is the law of conservation of angular momentum.)

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Chapter 13.4, Problem 42E

Chapter 13.4, Problem 45E

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Chapter 13 Solutions

EBK MULTIVARIABLE CALCULUS

Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Find the limit. 3. limt0(e3ti+t2sin2tj+cos2tk)Ch. 13.1 - Find the limit. 4. limt1(t2-tt-1i+t+8j+sintlntk)Ch. 13.1 - Find the limit. 5. limt1+t21t2,tan-1t,1e2tt Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

Ch. 13.1 - Prob. 11E Ch. 13.1 - Prob. 12E Ch. 13.1 - Prob. 13E Ch. 13.1 - Prob. 14E Ch. 13.1 - Prob. 15E Ch. 13.1 - Prob. 16E Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Prob. 18E Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Prob. 20E Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Prob. 27E Ch. 13.1 - Show that the curve with parametric equations x =...Ch. 13.1 - Prob. 29E Ch. 13.1 - Prob. 30E Ch. 13.1 - Prob. 31E Ch. 13.1 - At what points does the helix r(t) = sin t, cos t,...Ch. 13.1 - Graph the curve with parametric equations x = sin...Ch. 13.1 - Graph the curve with parametric equations x = (1 +...Ch. 13.1 - Prob. 40E Ch. 13.1 - Prob. 41E Ch. 13.1 - Prob. 42E Ch. 13.1 - Prob. 43E Ch. 13.1 - Prob. 44E Ch. 13.1 - Prob. 45E Ch. 13.1 - Prob. 46E Ch. 13.1 - If two objects travel through space along two...Ch. 13.1 - Prob. 50E Ch. 13.1 - Prob. 53E Ch. 13.2 - Prob. 1E Ch. 13.2 - Prob. 2E Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - Prob. 4E Ch. 13.2 - (a) Sketch the plane curve with the given vector...Ch. 13.2 - Prob. 6E Ch. 13.2 - Prob. 7E Ch. 13.2 - Prob. 8E Ch. 13.2 - Find the derivative of the vector function. 9....Ch. 13.2 - Prob. 10E Ch. 13.2 - Prob. 11E Ch. 13.2 - Prob. 12E Ch. 13.2 - Prob. 13E Ch. 13.2 - Prob. 14E Ch. 13.2 - Prob. 15E Ch. 13.2 - Prob. 16E Ch. 13.2 - Prob. 17E Ch. 13.2 - Prob. 18E Ch. 13.2 - Prob. 19E Ch. 13.2 - Prob. 20E Ch. 13.2 - If r(t) = t, t2, t3, find r'(t), T( 1), r"(t). and...Ch. 13.2 - Prob. 22E Ch. 13.2 - Prob. 23E Ch. 13.2 - Prob. 24E Ch. 13.2 - Prob. 25E Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find a vector equation for the tangent line to the...Ch. 13.2 - Find the point on the curve r(t) = 2 cos t, 2 sin...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - Find parametric equations tor the tangent line to...Ch. 13.2 - Prob. 32E Ch. 13.2 - Prob. 33E Ch. 13.2 - At what point do the curves r1(t) = t, 1 t, 3 +...Ch. 13.2 - Evaluate the integral. 35. 02(ti-t3j+3t5k)dt Ch. 13.2 - Prob. 36E Ch. 13.2 - Prob. 37E Ch. 13.2 - Evaluate the integral. 38....Ch. 13.2 - Evaluate the integral. 39....Ch. 13.2 - Evaluate the integral. 40. (te2ti+t1-tj+11-t2k)dt Ch. 13.2 - Find r(t) if r'(t) = 2t i + 3t2 j + t k and r(1) =...Ch. 13.2 - Prob. 42E Ch. 13.2 - Prove Formula 1 of Theorem 3.Ch. 13.2 - Prove Formula 3 of Theorem 3.Ch. 13.2 - Prob. 45E Ch. 13.2 - Prove Formula 6 of Theorem 3.Ch. 13.2 - Prob. 47E Ch. 13.2 - Prob. 48E Ch. 13.2 - Find f'(2), where f(t) = u(t) v(t), u(2) = 1, 2,...Ch. 13.2 - If r(t) = u(t) v(t), where u and v are the vector...Ch. 13.2 - If r(t) = a cos t + b sin t, where a and b are...Ch. 13.2 - If r is the vector function in Exercise 51, show...Ch. 13.2 - Show that if r is a vector function such that r''...Ch. 13.2 - Prob. 54E Ch. 13.2 - If r(t) 0, show that ddtr(t)=1r(t)r(t)r(t)....Ch. 13.2 - Prob. 56E Ch. 13.2 - Prob. 57E Ch. 13.2 - Prob. 58E Ch. 13.3 - Find the length of the curve. 1. r(t) =t, 3 cos t,...Ch. 13.3 - Find the length of the curve. 2. r(t)=2t,t2,13t3,...Ch. 13.3 - Prob. 3E Ch. 13.3 - Find the length of the curve. 4. r(t) =cos t i +...Ch. 13.3 - Find the length of the curve. 5. r(t) = i + t2 j +...Ch. 13.3 - Find the length of the curve. 6. r(t) = t2 i + 9t...Ch. 13.3 - Find the length of the curve correct to four...Ch. 13.3 - Find the length of the curve correct to four...Ch. 13.3 - Prob. 9E Ch. 13.3 - Graph the curve with parametric equations x = sin...Ch. 13.3 - Let C be the curve of intersection of the...Ch. 13.3 - Prob. 12E Ch. 13.3 - (a) Find the arc length function for the curve...Ch. 13.3 - Prob. 14E Ch. 13.3 - Suppose you start at the point (0, 0. 3) and move...Ch. 13.3 - Reparametrize the curve r(t)=(2t2+11)i+2tt2+1j...Ch. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - Prob. 18E Ch. 13.3 - (a) Find the unit tangent and unit normal vectors...Ch. 13.3 - Prob. 20E Ch. 13.3 - Use Theorem 10 to find the curvature. 21. r(t) =...Ch. 13.3 - Use Theorem 10 to find the curvature. 22. r(t) = t...Ch. 13.3 - Prob. 23E Ch. 13.3 - Prob. 24E Ch. 13.3 - Find the curvature of r(t) = t, t2, t3 at the...Ch. 13.3 - Graph the curve with parametric equations x = cos...Ch. 13.3 - Use Formula 11 to find the curvature. 27. y = x4...Ch. 13.3 - Prob. 28E Ch. 13.3 - Use Formula 11 to find the curvature. 27. y = x4...Ch. 13.3 - At what point does the curve have maximum...Ch. 13.3 - Prob. 31E Ch. 13.3 - Find an equation of a parabola that has curvature...Ch. 13.3 - (a) Is the curvature of the curve C shown in the...Ch. 13.3 - Two graphs, a and b, are shown. One is a curve y =...Ch. 13.3 - Prob. 39E Ch. 13.3 - Prob. 42E Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Consider the curvature at x = 0 for each member of...Ch. 13.3 - Prob. 47E Ch. 13.3 - Prob. 48E Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - At what point on the curve x = t3, y = 3t, z = t4...Ch. 13.3 - Find equations of the normal and osculating planes...Ch. 13.3 - Prob. 56E Ch. 13.3 - Prob. 58E Ch. 13.3 - Prob. 59E Ch. 13.3 - Show that the curvature of a plane curve is =...Ch. 13.3 - Prob. 62E Ch. 13.3 - Prob. 63E Ch. 13.3 - Show that the circular helix r(t) = a cos t, a sin...Ch. 13.3 - Prob. 65E Ch. 13.3 - Find the curvature and torsion of the curve x =...Ch. 13.3 - The DNA molecule has the shape of a double helix...Ch. 13.4 - The table gives coordinates of a particle moving...Ch. 13.4 - Prob. 3E Ch. 13.4 - Prob. 4E Ch. 13.4 - Prob. 5E Ch. 13.4 - Prob. 6E Ch. 13.4 - Prob. 7E Ch. 13.4 - Prob. 8E Ch. 13.4 - Prob. 9E Ch. 13.4 - Prob. 10E Ch. 13.4 - Prob. 11E Ch. 13.4 - Prob. 12E Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Prob. 14E Ch. 13.4 - Find the velocity and position vectors of a...Ch. 13.4 - Prob. 16E Ch. 13.4 - The position function of a particle is given by...Ch. 13.4 - Prob. 20E Ch. 13.4 - A force with magnitude 20 N acts directly upward...Ch. 13.4 - Show that if a particle moves with constant speed,...Ch. 13.4 - A projectile is fired with an initial speed of 200...Ch. 13.4 - Rework Exercise 23 if the projectile is fired from...Ch. 13.4 - A ball is thrown at an angle of 45 to the ground....Ch. 13.4 - A projectile is tired from a tank with initial...Ch. 13.4 - A rifle is fired with angle of elevation 36. What...Ch. 13.4 - Prob. 28E Ch. 13.4 - A medieval city has the shape of a square and is...Ch. 13.4 - Prob. 30E Ch. 13.4 - Prob. 31E Ch. 13.4 - A ball with mass 0.8 kg is thrown southward into...Ch. 13.4 - Prob. 34E Ch. 13.4 - Prob. 35E Ch. 13.4 - (a) If a particle moves along a straight line,...Ch. 13.4 - Prob. 37E Ch. 13.4 - Prob. 38E Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Prob. 40E Ch. 13.4 - Prob. 41E Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - If a particle with mass m moves with position...Ch. 13.4 - The position function of a spaceship is...Ch. 13.4 - A rocket burning its onboard fuel while moving...Ch. 13 - Prob. 1RCC Ch. 13 - What is the connection between vector functions...Ch. 13 - Prob. 3RCC Ch. 13 - Prob. 4RCC Ch. 13 - Prob. 5RCC Ch. 13 - Prob. 6RCC Ch. 13 - Prob. 7RCC Ch. 13 - Prob. 8RCC Ch. 13 - Prob. 9RCC Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Prob. 2RQ Ch. 13 - Prob. 3RQ Ch. 13 - Prob. 4RQ Ch. 13 - Prob. 5RQ Ch. 13 - Prob. 6RQ Ch. 13 - Prob. 7RQ Ch. 13 - Prob. 8RQ Ch. 13 - Prob. 9RQ Ch. 13 - Determine whether the statement is true or false....Ch. 13 - Prob. 11RQ Ch. 13 - Prob. 12RQ Ch. 13 - Prob. 13RQ Ch. 13 - Prob. 14RQ Ch. 13 - (a) Sketch the curve with vector function r(t) = t...Ch. 13 - Let r(t) = 2-t, (et 1)/t, ln(t + 1). (a) Find the...Ch. 13 - Prob. 3RE Ch. 13 - Find parametric equations for the tangent line to...Ch. 13 - If r(t) = t2 i + t cos t j + sin t k, evaluate...Ch. 13 - Let C be the curve with equations x = 2 t3 y = 2t...Ch. 13 - Use Simpsons Rule with n = 6 to estimate the...Ch. 13 - Prob. 8RE Ch. 13 - Prob. 9RE Ch. 13 - Prob. 10RE Ch. 13 - Prob. 11RE Ch. 13 - Prob. 12RE Ch. 13 - Find the curvature of the curve y = x4 at the...Ch. 13 - Find an equation of the osculating circle of the...Ch. 13 - Prob. 15RE Ch. 13 - The figure shows the curve C traced by a particle...Ch. 13 - A particle moves with position function r(t) = t...Ch. 13 - Find the velocity, speed, and acceleration of a...Ch. 13 - A particle starts at the origin with initial...Ch. 13 - An athlete throws a shot at an angle of 45 to the...Ch. 13 - A projectile is launched with an initial speed of...Ch. 13 - Prob. 22RE Ch. 13 - Prob. 23RE Ch. 13 - PROBLEM PLUS FIGURE FOR PROBLEM 1 1. A particle P...Ch. 13 - Prob. 2P Ch. 13 - A projectile is fired from the origin with angle...Ch. 13 - (a) A projectile it fired from the origin down an...Ch. 13 - Prob. 5P Ch. 13 - Prob. 6P Ch. 13 - If a projectile is fired with angle of elevation ...Ch. 13 - A cable has radius r and length L and is wound...Ch. 13 - Prob. 9P

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