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CentreCourseYear/ Trimester | : Centre for Foundation Studies (CFS): Foundation in Science: Year 1 / Trimester 1 | Session Course CodeCourse Title | : 201605: FHSC1014: Mechanics |

Tutorial 1: Introduction

1. What are the dimensions for the physical quantity expressed as , where m, g, d, T and l are the mass, acceleration due to gravity, distance, period and length respectively. A MT2 B ML2 C MLT2 D MLT–2 [Answer: B]

2. Suppose A = BC, where A has the dimension LM–1 and C has the dimension LT–1. What is the dimension of B? A TM–1 B L2T–1M–1 C TML–2 D L2TM–1 [Answer: A]

3. (a) State the dimension for (i) velocity (ii) force (b) Suppose an equation related with force F, viscosity ƞ, speed v, radius r is given by this formula , where k is a dimensionless constant; x, y, and z are the power to which ƞ, v and r. Given that the dimensions of viscosity ƞ is ML–1T–1. Use dimensional analysis to determine the value of x, y and z. [Answer: (a) (i) LT–1, (ii) MLT–2; (b) 1, 1, 1]

4. (a) Discuss briefly whether in general the method of dimensional checking can positively confirm that an equation is correct. (b) Newton’s law of universal gravitation is represented by

where F is the gravitational force, M and m are masses, and r is a length. Given that the dimensions of force is MLT–2. What are the dimensions and SI unit of the proportionality constant G? [Answer: (a) No; (b) L3M–1T–2, m3 kg–1 s–2]

5. The flow of a liquid with density ρ, round an obstacle of width b, changes from streamline to turbulence when the critical speed v is reached. If η is the viscosity of the liquid and given their relationship as below: where C is a dimensionless constant. Deduce the dimension of η. [Answer: ML–1T–1]

6. (a) The velocity of sound in medium of density ρ is given by the equation , where k is a constant. Deduce the dimension of k and state its unit in terms of SI base units. (b) Suppose the speed of a sail-boat is given by an equation , where k is a dimensionless constant, q is a constant with dimension, and l is the length of the boat. Find the dimension of the term q. [Answer: (a) ML–1T–2, kg m–1 s–2; (b) LT–2]

7. Thermal energy Q absorbed or released by an object is given by the expression: Q = mc where m = mass c = specific heat capacity = change in temperature of matter (a) What are the SI units of specific heat capacity, c? (b) If , what are the SI units of Y? [Answer: (a) J kg–1 K–1; (b) ]

8. The equation of de Broglie wavelength, λ is where h = Planck constant m = mass v = speed (a) What are the dimensions for Planck constant? (b) Write the base SI unit of Planck constant. [Answer: (a) L2MT–1; (b) m2 kg s–1]

9. The speed limit on an interstate highway is posted at 75 mi/h. What is this speed in (a) kilometers per hour (b) feet per second (c) meter per second? [Answer: (a) 121 km/h; (b) 110 ft/s; (c) 33.5 m/s]

10. Four lengths have been measured and recorded as follows: L1 = 3.427 m, L2 = 3.5 m, L3 = 0.333 m, L4 = 32.000 m (a) What is the uncertainty in each measurement? (b) What is the result if L1 + L2? (c) What is the result if L1 + L3? (d) What is the result if L1 – L3? (e) What is the result if L2 – L1? (f) What is the result if L1 L2? (g) What is the result if L4 L3? [Answer : (b) 6.9 m; (c) 3.760 m; (d) 3.094 m; (e) 0.1 m; (f) 12 m2; (g) 96.1]

11. The two hot air balloons in the following picture are 47.8 m and 61.2 m above the ground. A person in the left balloon observes that the right balloon is 28.5° above the horizontal.

What is the horizontal distance between the two balloons? [Answer: 24.7 m]

12. (a) The Cartesian coordinate of a point in xy-plane is (x, y) = (4.00 m, –3.50 m). Find the polar coordinate of this point. (b) Convert the polar coordinate (r, θ) = (5.00 m, 137°) into Cartesian coordinate. [Answer: (a) (5.32 m, 319°); (b) (–3.66 m, 3.41 m)]

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