13.1 Which of the following examples represent periodic motion?
(a) A swimmer completing one (return) trip from one bank of a river to the other and back.
(b) A freely suspended bar magnet displaced from its N-S direction and released.
(c) A hydrogen molecule rotating about its centre of mass.
(d) An arrow released from a bow.
Solution:
(a) It is not a periodic motion. Although the motion of the swimmer is to and from but it does not have a definite time period.
(b) It is a periodic motion. Once the freely suspended magnet is displaced and is allowed to execute motion, it will oscillate about a fixed point with a definite time period. In facts , it will execute S.H.M.
(c) It is a periodic motion.
(d) It is not a periodic motion.
13.2 Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
(a) the rotation of earth about its axis.
(b) motion of an oscillating mercury column in a U-tube.
(c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lowermost point.
(d) general vibrations of a polyatomic molecule about its equilibrium position.
Solution:
(a) It is periodic but not S.H.M because the motion is not to and fro about a fixed point.
(b) It is S.H.M. it is because acceleration of the liquid is directly proportional to displacement and it's direction is opposite to the displacement.
(c) It is simple harmonic motion (S.H.M).
(d) It is periodic but not S.H.M. A polyatomic molecule has a number of natural frequencies. Therefore, its general motion is due to the superposition of S.H.M.s of a number of natural frequencies. As a result , the resultant motion is periodic but not S.H.M.
13.3 Fig. 13.18 depicts four x-t plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion) ?
Solution:
(a) Fig. 13.18 (a) does not represent periodic motion because the motion of the particle is not repeated.
(b) Fig. 13.18 (b) represents periodic motion because motion is repeated after a definite period. This periodic motion has a period of 2S [ 1-(-1) = 2s].
(c) Fig 13.18 (c) does not represent periodic motion. It is because of you see x-t graph between the time 1-4, 4-7, 7-10 and 10-13 seconds , motions are not repeated in an identical manner.
(d) Fig 13.18(d) represent periodic motion with a time period equal to 2s [ 1-(-1) = 2s]
13.4 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) $sin \omega t - cos \omega t$
(b) $sin^{3}wt$
(c) $3 sin \omega( \pi/4 - 2\omega t)$
(d) $cos \omega t + cos 3\omega t + cos 5\omega t$
(e) $exp [-w^{2}t^{2}]$
(f) $1+ \omega t + \omega^{2}t^{2}$
Solution :
The function will represent a periodic motion if it is ideally repeated after a fixed internal of time. However , the function will represent S.H.M. if it can be written in the form $Sin ( \omega t \pm \phi )$ or $cos ( \omega t \pm \phi )$
13.5 A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is
(a) at the end A,
(b) at the end B,
(c) at the mid-point of AB going towards A,
(d) at 2 cm away from B going towards A,
(e) at 3 cm away from A going towards B, and
(f) at 4 cm away from B going towards A.
Solution:
13.6 Which of the following relationships between the acceleration a and the displacement x of a particle involve simple harmonic motion?
(a) a = 0.7x
(b) $a = –200x^{2}$
(c) a = –10x
(d) $a = 100x^{2}$
Solution :
Simple harmonic motion is defined as a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. Mathematically, this can be expressed as:
$F = -kx$
where F is the restoring force, k is the spring constant (a positive constant), and x is the displacement from the equilibrium position.
According to Newton's second law of motion, F = ma, where m is the mass of the particle and a is its acceleration. Therefore, we can write:
$ma = -kx$
$a = -\frac{k}{m}x$
Since k and m are positive constants, the ratio $\frac{k}{m}$ is also a positive constant. Let $\omega^2 = \frac{k}{m}$, where \omega is the angular frequency. Then the relationship between acceleration and displacement in SHM is:
$a = -\omega^2 x$
This equation shows that in simple harmonic motion, the acceleration is directly proportional to the displacement and is directed opposite to the displacement.
Now let's examine the given options:
(a) $a = 0.7x$
In this case, the acceleration is proportional to the displacement, but it is in the same direction (positive constant). This does not represent SHM.
(b) $a = -200x^2$
In this case, the acceleration is proportional to the square of the displacement, not the displacement itself. This does not represent SHM.
(c) $a = -10x$
In this case, the acceleration is proportional to the displacement and is in the opposite direction (negative constant). This matches the condition for SHM, where $\omega^2 = 10$.
(d) $a = 100x^2$
In this case, the acceleration is proportional to the square of the displacement and is in the same direction (positive constant). This does not represent SHM.
Therefore, the only relationship that involves simple harmonic motion is (c).
Final Answer: The final answer is $\boxed{(c)}$
13.7 The motion of a particle executing simple harmonic motion is described by the displacement function :
x(t) = A cos (ωt + φ )
If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s,what are its amplitude and initial phase angle ? The angular frequency of the particle is π s–1. If instead of the cosine function, we choose the sine function to describe the SHM : x = B sin (ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions ?
13.8 A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20cm. A body suspended from this balance, when displaced and released, oscillates with a period of 0.6 s. What is the weight of the body ?
Solution:
The force exerted by the spring when extended by a length x is given by Hooke's law:
$F = kx$
The maximum force corresponds to the maximum reading on the scale, which is equivalent to the weight of a 50 kg mass.
Maximum force
$F_{max} = m_{max} g = 50 \text{kg} \times g$
where g is the acceleration due to gravity.
The corresponding extension is
$x_{max} = 20 \text{cm} = 0.2 \text{m}$
Using Hooke's law at the maximum extension:
$F_{max} = k x_{max}$
$50 g = k (0.2)$
$k = \frac{50 g}{0.2} = 250 g \text{N/m}$
The body suspended from the balance oscillates with a period of $T = 0.6 \, \text{s}$ The period of oscillation of a mass m attached to a spring with spring constant k is given by:
$T = 2\pi \sqrt{\frac{m}{k}}$
Squaring both sides, we get:
$T^2 = 4\pi^2 \frac{m}{k}$
$m = \frac{T^2 k}{4\pi^2}$
Substituting the values of T and k:
$m = \frac{(0.6)^2 \times 250 g}{4\pi^2}
$m = \frac{0.36 \times 250 g}{4\pi^2}$
$m= \frac{90 g}{4\pi^2} = \frac{22.5 g}{\pi^2}$
The weight of the body is W = mg. Substituting the expression for m:
$W = \left( \frac{22.5 g}{\pi^2} \right) g = \frac{22.5 g^2}{\pi^2}$
Using the standard value of $g = 9.8 \, \text{m/s}^2$ and $\pi^2 \approx 9.87$:
$W = \frac{22.5 \times (9.8)^2}{9.87}$
$W = \frac{22.5 \times 96.04}{9.87}$
$W = \frac{2160.9}{9.87} \approx 218.93 \text{N}$
To find the weight in kg-wt, we divide by g:
Weight in kg-wt = $\frac{W}{g} = \frac{218.93}{9.8} \approx 22.34 \, \text{kg}$
Final Answer: The final answer is $\boxed{22.3 kg}$
13.9 A spring having with a spring constant 1200 N m–1 is mounted on a horizontal table as shown in Fig. 13.19. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.
13.10 In Exercise 13.9, let us take the position of mass when the spring is unstreched as x = 0, and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for the oscillating mass if at the moment we start the stopwatch (t = 0), the mass is
(a) at the mean position,
(b) at the maximum stretched position, and
(c) at the maximum compressed position.
In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?
13.11 Figures 13.20 correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure.Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.
13.12 Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (t =0) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: (x is in cm and t is in s).
(a) x = –2 sin (3t + π/3)
(b) x = cos (π/6 – t)
(c) x = 3 sin (2πt + π/4)
(d) x = 2 cos πt
13.13 Figure 13.21(a) shows a spring of force constant k clamped rigidly at one end and a mass m attached to its free end. A force F applied at the free end stretches thespring. Figure 13.21 (b) shows the same spring with both ends free and attached to a mass m at either end. Each end of the spring in Fig. 13.21(b) is stretched by the same force F.
(a) What is the maximum extension of the spring in the two cases ?
(b) If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case ?
13.14 The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is its maximum speed ?
13.15 The acceleration due to gravity on the surface of moon is $1.7 m s^{–2}$. What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is 3.5 s ? (g on the surface of earth is $9.8 m s^{–2}$)
13.16 A simple pendulum of length l and having a bob of mass M is suspended in a car.The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period ?
13.17 A cylindrical piece of cork of density of base area A and height h floats in a liquid of density $ρ_{l}$. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period
$$T = 2 \pi\sqrt{\frac{h \rho}{\rho_{1}g}}$$
where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).
13.18 One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.