HCI 07A Gravitation Solutions (Discussion Questions)

Gravitation Tutorial 7A Discussion Questions Suggested Solutions D1 F 100 = F400 O R s i m p l y g100 = g400 𝐺 ሺଵ଴଴ሻ௠ ௫మ ൌ𝐺 ሺସ଴଴ሻ௠ ሺଵ଴.଴ି௫ሻమ 𝐺 ሺଵ଴଴ሻ ௫మ ൌ𝐺 ሺସ଴଴ሻ ሺଵ଴.଴ି௫ሻమ x = 3.33 m D2 B Vector sum of gravitational field strengths at B due to each of the 3 bodies = 0 at the neutral point Gravitational field strength at a point due to a mass is propor tional to the mass and inversely proportional to the square of the distance between the mass and the point. Consider Earth and Moon’s intera ction first, the point where th e resultant gravitational field strength is zero is nearer to the smaller mass (refer to D1 whe re g = 0 is nearer the 100 kg mass). Then consider Moon and Sun’s interaction, which the neut ral point will be shifted nearer to the Sun, but it will be still nearer to the smaller mass. Good to know:   Mass of the Earth = 5.97 x 10 24 kg,  mass of the Moon = 7.35 x 1022 kg,   mass of the Sun = 1.99 x 1030 kg  D3 2 2 98 1E E E MgG . m sR  (a) 2 22 1 new E EE n e w g Rg Rg R  2 2 2 1 24 524 E new E E E Rgg g . m s(R)   m  F400 F100 x 10.0 - x 100 kg  400 kg  NOTE: The concept of finding the point where the resultant force  acting on a mass due to the two masses equals zero is equivalent  to finding the point where the resultant gravitational field  strength due to the two masses equals zero.  The concept of using ratio is a common method to  solve many physics problems.

(b)     3 22 4 43 3 GRGMgG R RR gR     Same radius but twice the density  g’ = 19.6 m s-2 (c) Half radius but twice the density  g’ = 9.81 m s-2 D4 B 3 22 4 43 3 65 1 8 3.613 5 EE E E E MM M M M GrGMgG r g r rr gr r r gr r r                D5 Apparent weight of zero means no force on its support  normal force by ground on it = 0 The entire gravitational force provides the centripetal force.  cWNm a (since N = 0)    2 2 cWm a mg mr T Taking g at the surface to be 9.81 m s-2, and the given radius of Earth, T = 5100 s (2 s.f.) D6 (2009 P3 Q5) (a)(i) The gravitational field strength at a point is the gravitational force per unit mass acting on a small test mass placed at that point. (ii) Newton’s Law of Gravitation states that every point mass (or particle) attracts every other point mass (or particle) with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between t hem. Thus 2 GMmF R 2 2 Since , GMmmg R GMg m R Fg   (b)(i) Average density =  30 17 3 34 total mass 5.2 10 2.53 104total volume 1.7 103 kg m     The concept of using ratio again similar to D3.

(ii) Gravitational force acts toward the centre of the neutron star; all the particles are pulled inwards. The outer layers compress the inner layers, resulting in increased density towards the centre. (c)(i)     11