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A particle of mass m moves under the central repulsive force ((mb)/r^3 ) and is initially moving at a distance ′a′ from the origin of a force with velocity ′v′ at right angle to ′a′. show that rcos pθ=a where p =(b/(a^2 v^2 ))+1.

$${A}\:{particle}\:{of}\:{mass}\:{m}\:{moves}\:{under}\:{the}\:{central} \\ $$$${repulsive}\:{force}\:\frac{{mb}}{{r}^{\mathrm{3}} }\:\:{and}\:{is}\:{initially}\:{moving} \\ $$$${at}\:{a}\:{distance}\:'{a}'\:\:{from}\:{the}\:{origin}\:{of}\:\:{a}\:{force} \\ $$$${with}\:{velocity}\:\:'{v}'\:{at}\:{right}\:{angle}\:{to}\:\:'{a}'. \\ $$$${show}\:{that}\:\:\: \\ $$$$\:\:\:\:\:{r}\mathrm{cos}\:{p}\theta={a}\:\:{where}\:{p}\:=\frac{{b}}{{a}^{\mathrm{2}} {v}^{\mathrm{2}} }+\mathrm{1}. \\ $$$$ \\ $$

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A massless spring with a spring constant K=10Nm^(−1) is suspended from rigid support carries a mass m=100g at it lower end the system is subjected to resistive force −pv.it observed that the system oscillates and its energy to one third of it initial value in one and half minutes calculates the value of p

$$\mathrm{A}\:\mathrm{massless}\:\mathrm{spring}\:\mathrm{with}\:\mathrm{a}\:\mathrm{spring}\:\mathrm{constant} \\ $$$$\mathrm{K}=\mathrm{10Nm}^{−\mathrm{1}} \:\mathrm{is}\:\mathrm{suspended}\:\mathrm{from}\:\mathrm{rigid}\:\mathrm{support} \\ $$$$\mathrm{carries}\:\mathrm{a}\:\mathrm{mass}\:\mathrm{m}=\mathrm{100g}\:\mathrm{at}\:\mathrm{it}\:\mathrm{lower}\:\mathrm{end} \\ $$$$\mathrm{the}\:\mathrm{system}\:\mathrm{is}\:\mathrm{subjected}\:\mathrm{to}\:\mathrm{resistive}\:\mathrm{force} \\ $$$$−\mathrm{pv}.\mathrm{it}\:\mathrm{observed}\:\mathrm{that}\:\mathrm{the}\:\mathrm{system}\:\mathrm{oscillates} \\ $$$$\mathrm{and}\:\mathrm{its}\:\mathrm{energy}\:\mathrm{to}\:\mathrm{one}\:\mathrm{third}\:\mathrm{of}\:\mathrm{it}\:\mathrm{initial}\:\mathrm{value} \\ $$$$\mathrm{in}\:\mathrm{one}\:\mathrm{and}\:\mathrm{half}\:\mathrm{minutes}\:\mathrm{calculates}\:\mathrm{the}\: \\ $$$$\mathrm{value}\:\mathrm{of}\:\:\mathrm{p} \\ $$

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A projectile of mass M explodes at thee highst point of its trajectory when it hase vlocity . The horizontal distance travelede btween launch and explosion is x_0 . Two fragments are produced with initiale velocitis parallel to the ground. They thenfollow their trajectories until they hitt he ground. The fragment of mass m_1 retuns exactly to the launch point of thei orginal projectile (of mass M) while thee othr fragment of mass m_2 hits the grounda t a distance D from this point. Disregardn iteraction with air and assume that massa ws conserved in the explosion (m_1 +m_2 =M) Determine the magnitude of the velocity of fragment 2 just before it hits theground. (a) ((gx_0 )/v) (b)(√((25)/9))v (c) (√(((25)/9)v^2 +(((gx_0 )/5))2)) (d)(√((5/3)x_0 v^2 +(((gx_0 )/v))2))

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