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1PP Examination Autumn 2002 Examiner Dr. S. J. Sweeney Note: All vector quantities are BOLD FACED and UNDERLINED Unit vectors are denoted as iˆ , ĵ , k̂ SECTION A: ANSWER ALL QUESTIONS 1. If the radius of the Earth is R, and the acceleration due to gravity at the surface is g, what is the acceleration due to gravity at a height 2 1 R above the Earths surface? (a) 0.5g (b) 2g (c) g (d) (e) 2. g 2 1 g One object of mass 2kg moves along the x-axis with speed -2vx. Another object of mass 4kg moves along the x-axis with speed +vx. What is the centreof-mass velocity for this system? (a) -2vx (b) 2vx (c) vx (d) -vx (e) 0 3. Calculate the determinant: 1 2 1 0 2 0 2 2 2 (a) -2 (b) 0 (c) 2 (d) 4 (e) 8 4. Two forces of ( iˆ + ĵ + k̂ ) N and (2 iˆ -2 ĵ - k̂ ) N act on an object. The unit vector in the direction of the resultant force is: (a) (3 iˆ - ĵ ) N (b) (c) (d) (e) 5. 10 N 1 10 1 2 (3 iˆ - ĵ ) N (3 iˆ - ĵ ) N 1 10 (-3 iˆ + ĵ ) N What is the vector product a b where a = ( iˆ + ĵ - k̂ ) and b = (3 iˆ -2 ĵ + k̂ ): (a) 3 iˆ +2 ĵ -5 k̂ (b) - iˆ -4 ĵ -5 k̂ (c) - iˆ +4 ĵ -5 k̂ (d) 3 iˆ -4 ĵ -5 k̂ (e) 0 6. A photon has an energy of 1000meV. This is equivalent to: (a) 1.6 x 1016 J (b) 1.6 x 10-16 J (c) 1.6 x 1019 J (d) 1.6 J (e) 1.6 x 10-19 J 7. An object moves from iˆ m to ( iˆ + ĵ - k̂ ) m whilst being acted upon by a force ( iˆ + ĵ + k̂ ) N. What is the work done by the force on the object? (a) 0 J (b) 2 J (c) ( iˆ + ĵ ) J (d) 1 J (e) 8. 5 J The extension, d, of a spring which obeys Hooke's law is proportional to the tension, F, which has produced it. Therefore, F=kd. The dimensions of the constant, k, in terms of mass [M], length [L] and time, [T] are: (a) [M][L]2[T]-2 (b) [M][T]-2 (c) k is a dimensionless constant (d) [M]-1[T]2 (e) [L][T]-1 9. Prior to the metric system, the French unit of length was the toise, where 1 toise = 1949 mm. The toise was divided into 6 Paris feet, each of which was divided into 12 pouces, that were further subdivided into 12 lignes. How many cubic lignes occupy the same volume as 1 cubic centimetre? 10. (a) 8.71x10-2 (b) 0.871 (c) 8.71 (d) 87.1 (e) 871 Which of the following is not an SI unit? (a) volt (b) kelvin (c) mol (d) metre (e) candela 11. A particle moves in a circle with constant angular velocity, ω . If the origin of co-ordinates is at the centre of the circle, which one of the following statements concerning the velocity, v , and position, r , of the object is false? (a) r is constant (b) r is constant (c) v is constant (d) v ω r (e) v r 12. If a nucleus is represented by an object the size of a 1p coin, indicate the distance between two 1p coins that would best correspond to the distance between two nuclei in a crystal. Assume size of nucleus is 10-15m and typical spacing of nuclei in a crystal is 10-10m. 13. (a) 1 cm (b) 1m (c) 1 km (d) 1000 km (e) 106 km A particle has a velocity given by v = 3t2 iˆ - ĵ ms-1 The position of the particle as a function of time, given that the particle was at the origin at t = 2s, is 14. (a) t3-t m (b) t3 iˆ m (c) (t3-8) iˆ + (2-t) ĵ m (d) t3 iˆ - t ĵ m (e) (t3-8) iˆ m Which of the following statements about an elastic collision between two objects is false? (a) Some kinetic energy may be converted to heat or sound. (b) The two objects cannot bind to form a composite body. (c) Total energy is conserved. (d) The two objects exert equal and opposite forces on each another. (e) The centre-of-mass velocity remains constant. 15. One light-year (ly) is a unit of length corresponding to the distance that light travels in one Earth year. To what order of magnitude is the distance from the Earth to the moon in light years (assume that the speed of light, c=3x108ms-1 and that the moon is 4x105km from Earth). (a) 10-14 ly (b) 10-11 ly (c) 10-8 ly (d) 10-5 ly (e) 10-2 ly (i) SECTION B: ANSWER TWO QUESTIONS 1. (a) The work done by a conservative force, F , on an object moving from P1 to P2 is given by the line integral: P2 W F dr P1 (i) (ii) (iii) Explain what is meant by a conservative force. Give one example of a conservative force and one example of a nonconservative force. Briefly explain why they are considered to be conservative or non-conservative. If F is not a function of position, show that this equation simplifies to W Fr What does the vector r represent? 12 marks (b) In 1969 the Apollo 11 lunar module (Eagle) successfully landed on the moon. After completing the mission, the Eagle module then blasted-off the surface to re-join the command module 15km above the moon's surface. (i) (ii) (iii) (iv) Assuming that the mass of the Eagle module is constant during its ascent and equal to 5000kg (of which 2700kg is fuel) and that the acceleration due to gravity on the moon, g=1.6ms-1, is assumed to be constant during the ascent, calculate the work required to lift Eagle to the command module. By how much has the gravitational potential energy of Eagle increased upon reaching the command module? Given that the radius of the moon, Rmoon=1738km, what is the percentage change in g on going from the surface to 15km above the surface? Is the value calculated for the work done in part (i) likely to be an over-estimate or under-estimate? Give reasons. 8 marks 2. (a) Explain what is meant by, (i) (ii) (iii) (iv) (v) (vi) (vii) the component of a vector the magnitude of a vector the resultant of a set of vectors a unit vector If a 5iˆ 3 ˆj 7kˆ and b 3iˆ ˆj 4kˆ , calculate the unit vector in the direction of a b . What is the smallest angle between vectors a and b ? Calculate the vector product, a b . 10 marks (b) A particle with charge q and mass m has a velocity v in a region containing a magnetic field B . The force experienced by the particle is given by F qv B If the magnetic field has a component in the x direction only, show that the acceleration of the particle is given by a qB ˆ vz j v y kˆ m where vz and vy are the z and y components of v respectively. (c) A particle of mass, m, has a velocity given by the vector equation, v Aiˆ C sin tˆj costkˆ where A, C and are constants. Determine the acceleration of this particle. (d) Show that the particle in part (c) has an acceleration and velocity appropriate to a particle moving in a region containing a uniform magnetic field in the x direction, as described in part (b). Hence obtain an expression for . 10 marks 3. (a) A disc of radius, r, thickness, y, and uniform density, , rotates with an d angular velocity, , where . dt (i) By considering a sector of the disc, as shown below, develop an expression for the magnitude of the linear velocity, v, in terms of and r. v d dx r (ii) Define the moment of inertia. What does it physically represent? (iii) Given that the moment of inertia, I, of a disc is I mr 2 , show that the 2 angular momentum, L, of the disc, is given by L 12 r 4 y (iv) For a disc of diameter 1m, thickness 10cm and density 10g/cm3, calculate its angular momentum if it rotates at 60 revolutions per minute. 11 marks (b) A audio compact disc (CD) consists of an approximately uniform disc of diameter 120mm, centred on a hole of diameter 15mm. Data is recorded on the disc from a distance of 24mm from the disc centre to a distance of 58mm from the disc centre forming a helical track. During playback, the CD is rotated such that the data is read at a constant linear velocity, v, of 1.3ms-1. (i) (ii) (iii) Sketch and label a graph showing how the angular velocity, , varies as a function of the radial distance r such that v is kept constant. Show the minimum and maximum angular velocities on the graph. Using the information given above, calculate the total length of the helical data track for a standard 75 minute audio CD. To the nearest order of magnitude, estimate the width (pitch) of the data track on the CD. 9 marks