Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (ω is any positive constant):
(a) sin ωt - cos wt
(b) sin3 ωt
(c) 3 cos (π/4 - 2ωt)
(d) cos ωt + cos 3ωt + cos 5ωt
(e) exp (-ω2t2)
(a) SHM
The given function is:
sin ωt - cos wt
= underroot 2 [1/underroot 2 sin ωt - 1/underroot 2 cos ωt]
= underroot 2 [ sin ωt x cos π/4 - cos ωtx cos π/4]
= underroot 2 ( ωt - π/4)
This function represents SHM as it can be written in the form:
a sin ( ωt + ø )
Its period is: 2π/ω
(b) Periodic, but not SHM
The given function is:
sin3ωt = ¼ [3sin ωt –sin 3ωt]
Even though the two sin ωt represent simple harmonic motions respectively, but they are periodic because superposition of two SIMPLE HARMONIC MOTION is not simple harmonic.
(c) 3 cos (π/4 – 2ωt) = 3 cos (2ωt – π/4)
As it can be written as : a sin ( ωt + Φ) , it represents SIMPLE HARMONIC MOTION
Its period is : π/ω
(d) In cos ωt + cos 3ωt + cos 5ωt, each cosine function represents SIMPLE HARMONIC MOTION, but the super position of SIMPLE HARMONIC MOTION gives periodic.
(e) As it is an exponential function, it is non periodic as it does not repeat itself.
( f ) 1 + ωt + ω2 t2 is non periodic.
State the number of significant figures in the following:
(a) 0.007 m2
(b) 2.64 x 1024 kg
(c) 0.2370 g cm-3
(d) 6.320 J
(e) 6.032 N m-2
(f) 0.0006032 m2
Fill in the blanks by suitable conversion of units:
(a) 1 kg m2s–2= ....g cm2 s–2
(b) 1 m =..... ly
(c) 3.0 m s–2=.... km h–2
(d) G = 6.67 × 10–11 N m2 (kg)–2=.... (cm)3s–2 g–1.
A physical quantity P is related to four observables a, b, c and d as follows :
The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2%, respectively. What is the percentage error in the quantity P ? If the value of P calculated using the above relation turns out to be 3.763, to what value should you round off the result ?
Rain is falling vertically with a speed of 30 m s–1. A woman rides a bicycle with a speed of 10 m s–1 in the north to south direction. What is the direction in which she should hold her umbrella?
The mass of a box measured by a grocer's balance is 2.300 kg. Two gold pieces of masses 20.15 g and 20.17 g are added to the box. What is
(a) the total mass of the box,
(b) the difference in the masses of the pieces to correct significant figures?
On an open ground, a motorist follows a track that turns to his left by an angle of 60° after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
What amount of heat must be supplied to 2.0 x 10-2 kg of nitrogen (at room temperature) to raise its temperature by 45 °C at constant pressure? (Molecular mass of N2 = 28; R = 8.3 J mol-1 K-1.)
A transverse harmonic wave on a string is described by
y(x,t) = 3.0 sin [36t + 0.018x + π /4]
Where x and y are in cm and t in s. The positive direction of x is from left to right.
(a) Is this a travelling wave or a stationary wave? If it is travelling, what are the speed and direction of its propagation?
(b) What are its amplitude and frequency?
(c) What is the initial phase at the origin?
(d) What is the least distance between two successive crests in the wave?
Fill in the blanks
(a) The volume of a cube of side 1 cm is equal to.....m3
(b) The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to ... (mm)2
(c) A vehicle moving with a speed of 18 km h–1covers....m in 1 s
(d) The relative density of lead is 11.3. Its density is ....g cm–3 or . ...kg m–3.
Read each statement below carefully and state, with reasons and examples, if it is true or false:
A scalar quantity is one that
(a) is conserved in a process
(b) can never take negative values
(c) must be dimensionless
(d) does not vary from one point to another in space
(e) has the same value for observers with different orientations of axes
Estimate the average thermal energy of a helium atom at
(i) room temperature (27 °C),
(ii) the temperature on the surface of the Sun (6000 K),
(iii) the temperature of 10 million Kelvin (the typical core temperature in the case of a star).
A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone?
Fill in the blanks by suitable conversion of units:
(a) 1 kg m2s–2= ....g cm2 s–2
(b) 1 m =..... ly
(c) 3.0 m s–2=.... km h–2
(d) G = 6.67 × 10–11 N m2 (kg)–2=.... (cm)3s–2 g–1.
The principle of ‘parallax’ in section 2.3.1 is used in the determination of distances of very distant stars. The baseline AB is the line joining the Earth’s two locations six months apart in its orbit around the Sun. That is, the baseline is about the diameter of the Earth’s orbit ≈ 3 × 1011m. However, even the nearest stars are so distant that with such a long baseline, they show parallax only of the order of 1” (second) of arc or so. A parsec is a convenient unit of length on the astronomical scale. It is the distance of an object that will show a parallax of 1” (second) of arc from opposite ends of a baseline equal to the distance from the Earth to the Sun. How much is a parsec in terms of metres ?
Fill in the blanks using the word(s) from the list appended with each statement:
(a) Surface tension of liquids generally . . . with temperatures (increases / decreases)
(b) Viscosity of gases. .. with temperature, whereas viscosity of liquids . . . with temperature (increases / decreases)
(c) For solids with elastic modulus of rigidity, the shearing force is proportional to . . . , while for fluids it is proportional to . .. (shear strain / rate of shear strain)
(d) For a fluid in a steady flow, the increase in flow speed at a constriction follows (conservation of mass / Bernoulli's principle)
(e) For the model of a plane in a wind tunnel, turbulence occurs at a ... speed for turbulence for an actual plane (greater / smaller)
A block of mass 25 kg is raised by a 50 kg man in two different ways as shown in Fig. 5.19. What is the action on the floor by the man in the two cases? If the floor yields to a normal force of 700 N, which mode should the man adopt to lift the block without the floor yielding?
In deriving Bernoulli's equation, we equated the work done on the fluid in the tube to its change in the potential and kinetic energy. (a) What is the largest average velocity of blood flow in an artery of diameter 2 x 10-3 m if the flow must remain laminar? (b) Do the dissipative forces become more important as the fluid velocity increases? Discuss qualitatively.
What amount of heat must be supplied to 2.0 x 10-2 kg of nitrogen (at room temperature) to raise its temperature by 45 °C at constant pressure? (Molecular mass of N2 = 28; R = 8.3 J mol-1 K-1.)