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Question 3 A particle of mass m moves in a plane with origin O, and its plane polar coordinates are (r(t), 0(t)). It is subject to a force directed at O of magnitude mF(r). a. Show that the radial and transverse components after applying Newton's law reduce to: i. F-10²=-F(r), ii. 7-²0 = h, where h is a positive constant (and, in the usual notation, a dot over a variable denotes its time derivative). b. Make the substitution that u=, and use (ii), iii. to show that iv. and deduce that =-h h² (du + u) = u^²F(u−¹). c. Solve (b.) for the case F(r) = r2, where is a positive constant. where e is a constant. du d. Given that there is a point on the path at which = 0, show that in the case when F(r) = μr-2, the equation of the path can be taken to be 1 ===(¹- (1 + e cos 0),

Question 3 A particle of mass m moves in a plane with origin O, and its plane polar coordinates are (r(t), 0(t)). It is subject to a force directed at O of magnitude mF(r). a. Show that the radial and transverse components after applying Newton's law reduce to: i. F-10²=-F(r), ii. 7-²0 = h, where h is a positive constant (and, in the usual notation, a dot over a variable denotes its time derivative). b. Make the substitution that u=, and use (ii), iii. to show that iv. and deduce that =-h h² (du + u) = u^²F(u−¹). c. Solve (b.) for the case F(r) = r2, where is a positive constant. where e is a constant. du d. Given that there is a point on the path at which = 0, show that in the case when F(r) = μr-2, the equation of the path can be taken to be 1 ===(¹- (1 + e cos 0),

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Question 3 A particle of mass m moves in a plane with origin O, and its plane polar coordinates are (r(t), 0(t)). It is subject to a force directed at O of magnitude mF(r). a. Show that the radial and transverse components after applying Newton's law reduce to: i. F-10²=-F(r), ii. 1²0 = h, where h is a positive constant (and, in the usual notation, a dot over a variable denotes its time derivative). b. Make the substitution that u=1, and use (ii), iii. to show that iv. and deduce that =-h where e is a constant. h² (du + u) = u^²F(u−¹). c. Solve (b.) for the case F(r) = r2, where is a positive constant. d. Given that there is a point on the path at which = 0, show that in the case when F(r) = μr-2, the equation of the path can be taken to be du |=-=(1- (1 + e cos 0),
Transcribed Image Text:Question 3 A particle of mass m moves in a plane with origin O, and its plane polar coordinates are (r(t), 0(t)). It is subject to a force directed at O of magnitude mF(r). a. Show that the radial and transverse components after applying Newton's law reduce to: i. F-10²=-F(r), ii. 1²0 = h, where h is a positive constant (and, in the usual notation, a dot over a variable denotes its time derivative). b. Make the substitution that u=1, and use (ii), iii. to show that iv. and deduce that =-h where e is a constant. h² (du + u) = u^²F(u−¹). c. Solve (b.) for the case F(r) = r2, where is a positive constant. d. Given that there is a point on the path at which = 0, show that in the case when F(r) = μr-2, the equation of the path can be taken to be du |=-=(1- (1 + e cos 0),
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