Questions on Gravity and Orbits MS 1. Using the usual symbols write down an equation for (i) Newton s law of gravitation M1M F G R (ii) Coulomb s law Q1Q F K R State one difference and one similarity between gravitational and electric fields. Difference Gravitational fields are attractive but electric fields can be attractive or repulsive ( marks) Similarity Both have an range A speck of dust has a mass of 1.0 10-18 kg and carries a charge equal to that of one electron. Near to the arth s surface it experiences a uniform downward electric field of strength 100 N C -1 and a uniform gravitational field of strength 9.8 N kg -1. Draw a free-body force diagram for the speck of dust. Label the forces clearly. lectric force ( marks) Gravitational force Calculate the magnitude and direction of the resultant force on the speck of dust. lectric force 100 N C -1 1.6 10-19 C 1.6 10-17 N Weight 9.8 10-18 N Net force is upward Force 6. 10-18 N (6 marks) [Total 10 marks] PhysicsAndMathsTutor.com 1
. The diagram (not to scale) shows a satellite of mass m, in circular orbit at speed v s around the arth, mass M. The satellite is at a height h above the arth s surface and the radius of the arth is R. υ s Satellite h arth R Using the symbols above write down an expression for the centripetal force needed to maintain the satellite in this orbit. m v s s F () R + h Write down an expression for the gravitational field strength in the region of the satellite. GM g ( R + h) () State an appropriate unit for this quantity. N kg -1 Use your two expressions to show that the greater the height of the satellite above the arth, the smaller will be its orbital speed. m v GM m s s R + h s ( R + h) v s GM R + h Greater h Þ smaller v s since G, M constant xplain why, if a satellite slows down in its orbit, it nevertheless gradually spirals in towards the arth s surface. GM ms m v s s As it slows > ( R + h) R + h The spare gravitational force not needed to provide the centripetal acceleration pulls the satellite nearer to the arth ( marks) (3 marks) (3 marks) ( marks) [Total 10 marks] PhysicsAndMathsTutor.com
3. Forces (i) F GM m/r 1 (ii) F GM M m/r 1 Distance R M M m GM m GM mm R r OR 1 R M OR r M m R r 81 R 7 1 (3.9 10 m) R 3.5 10 8 m vidence that equating forces has occurred 1 Correct substitution 1 Correct answer 1 [5] 4. Gravitational attraction of arth on Moon Gm1m Use S, ie r Orbital speed of the Moon GMm 1 ( 60R) mυ r GMm r Use of r 60R, ie 60 6.4 10 6 11 6.67 10 Nm kg 6 10 kg Rearrangement ie υ 6 60 6.4 10 m 100 m s 1 4 Orbit period πr π Time /ω υ T 6 π 60 6.4 10 Calculation: 1 100 m s Divide by 3600 4 Using 100m/s: (7 7.4) days Using 1000m/s: (7.8 8) days 4 4 [9] PhysicsAndMathsTutor.com 3
5. Word equation: Force proportional to product of masses and inversely proportional to (distance / separation) squared [No force 0/] OR G mass 1 mass F (distance ) [or (separation) instead of bottom line] Calculation of force: From Newton s law OR idea that force weight mg planet 6.67 10 F 11 N m kg (3.40 10 6.4 10 [Substitution in correct equation only] OR 6 ) m 3 kg 1 kg 3 G 6.4 10 kg g mars 6 (3.4 10 ) m 3.7 N Smaller 3 xplanation of reasoning: g is less, but ρ is similar/same [so R is less] [ nd mark is consequential on first mark] [7] 6. Base units of G: kg 1 m 3 s quation homogeneity: Correct substitution of units of G, r 3 (m 3 ), M (kg) 1 leading to S and linked to T [Allow e.c.f. of their base unit answer into substitution mark] 3 Use of relationship to find mass of the arth Any two from: Adding 0 000 + 6400 Converting km to m h to s ( 43 00) Answer M 5.8 (4) 10 4 kg 1 [6] PhysicsAndMathsTutor.com 4
7. xpression for gravitational force F GMm/r 1 Derived expression Reasoning step must be clear, e,g, mg GMm/r so g GM/r Sun's gravitational field strength g 6.67 10 11 N m kg - 1.99 10 30 kg / (1.50 10 11 m) 5.9 10-3 (N kg -1 ) [no u.e.] Diagram (i) Jupiter marked closest to arth (ii) (Labelled) arrows towards Jupiter and Sun (radially) Maximum percentage change 3. 10-7 / 5.9 10-3 100% 0.005% 1 Maximum value of the ratio Use of g M/r Hence g Venus/ g jupiter 15 /400 0.56 [OR 9/16] Comment.g. reference to % change caused by Jupiter (combined with effect caused by Venus) OR g of (all) planets is very small compared with g of Sun 1 [11] 8. Show that: F GMm/r 6.9 10 4 (N) Calculation: a F/M 3.1 10 6 m s 1 xplanation: Planet exerts gravitational force on star Planet revolves around star, so direction of force changes with time Diagram showing force (or effect of force on star due to planet) 3 PhysicsAndMathsTutor.com 5
Speed of star: Using υ π r/ T and a υ / r r υt/π so a πυ / T so υ at / π 3.1 10 6 9. 10 7 π [allow ecf for a] 45.4 m s 1 3 Calculation: λ λ υ/c 656 10 9 45 / 3.0 10 8 9.8 10 14 m [Accept λ for maximum marks] [11] 9. Formula F GMm/r 1 Show that tan θ MR / M e r Horizontally Tsinθ F mountain and vertically Tcosθ mg [OR vector diagram showing forces and θ] Dividing equations [OR from vector diag.]: tanθ F mountain mg and F mountain GMm r and mg GM e m R so tan θ GMm/r GM e m/r 4 Value for gravitational constant, G Volume 4/3πR 3 4/3π (6.4 10 6 m) 3 1.1 10 1 (m 3 ) M e Vρ 1.1 10 1 m 3 4.5 10 3 kg m 3 4.9 10 4 (kg) G gr M e 3gR 4πR 3 ρ 3g 4πRρ 3 9.8 m s 6 4π 6.4 10 m 4.5 10 kg m 3 3 8.1 10 11 N m kg 3 Reason for inaccuracy Any one from: Maskelyne s density is incorrect arth/mountain not uniform density (centre of) mass of mountain not known mountain is not spherical difficult to determine vertical / measure very small θ arth not a perfect sphere / point mass 1 arth s core Much denser than mountain 1 [10] PhysicsAndMathsTutor.com 6
10. Minimum mass for comet Volume 4/3π (9 10 3 / ) 3 m 3 ( 3.8 10 11 m 3 ) Use of mass density volume 500 kg m 3 4/3 π (9 10 3 / ) 3 m 3 1.91 10 14 kg 3 Jupiter s gravitational field strength g GM/r 6.670 10 11 N m kg, 1.8987 10 7 kg / (96 009 10 3 ) 13.743 N kg 1 3 xplanation Any two from: Jupiter s force/field strength different on the two sides of the comet difference sufficient to pull comet apart Jupiter s force larger than (cohesive) force between particles of comet Max Difference and similarity between gravitational and electric fields Difference: e.g. gravitational fields only attractive, electrical can be attractive or repulsive, gravitational fields due to mass, electrical due to charge, cannot shield g-field, can shield field Similarity: e.g. follow equivalent mathematical formulae, both obey inverse square law [10] 11. Definitions: An electric field is a region where charged objects experience a force ( F/Q) 1 A gravitational field is a region where masses experience a force(g F/m) 1 Similarities: Both fields obey an inverse square law OR inverse square equations quoted Both fields are radial for point objects / spherical distributions Both fields have an infinite range / field strength approaches zero a long way from source max Differences: lectric forces can be attractive or repulsive but gravitational forces are always attractive lectric forces are (much) stronger than gravitational forces OR comparison of size of coupling constants in the two force equations lectric forces only act on charged particles but gravitational forces act on all matter lectric forces can be shielded (e.g. by use of a Faraday cage) but gravitational forces cannot max 3 [6] PhysicsAndMathsTutor.com 7
1. Magnitude of gravitational force on Cassini F GMm/r 1 xpression g F/m so g GM/r Maximum acceleration Appreciation that acceleration g-field Addition of orbital height to radius of Venus g G 4.87 10 4 kg / (6384 10 3 ) 7.97 m s 3 ffect of acceleration on velocity of Cassini Any from: Acceleration is at right angles to direction of motion Speed unchanged (Velocity changed since) direction changed Max [8] PhysicsAndMathsTutor.com 8