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Page No 360: - Chapter 14 Oscillations class 11 ncert solutions Physics - SaraNextGen [2024]


Question 14.11:

Figures 14.29 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.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/8096/Chapter%2014_html_m72f3bc6.jpg

Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.

Answer:

(a) Time period, = 2 s

Amplitude, A = 3 cm

At time, = 0, the radius vector OP makes an angle https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/8096/Chapter%2014_html_m4c4df2e.gif  with the positive x-axis, i.e., phase anglehttps://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/8096/Chapter%2014_html_m15a1bbda.gif

Therefore, the equation of simple harmonic motion for the x-projection of OP, at time t, is given by the displacement equation:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/8096/Chapter%2014_html_mebd02ec.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/8096/Chapter%2014_html_m39d64664.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/8096/Chapter%2014_html_m16d59b74.gif

(b) Time period, = 4 s

Amplitude, a = 2 m

At time t = 0, OP makes an angle π with the x-axis, in the anticlockwise direction. Hence, phase angle, Φ = + π

Therefore, the equation of simple harmonic motion for the x-projection of OP, at time t, is given as:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/8096/Chapter%2014_html_410a65f0.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/8096/Chapter%2014_html_m4a149359.gif

Question 14.12:

Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (= 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: (is in cm and is in s).

(a) = –2 sin (3+ π/3)

(b) = cos (π/6 – t)

(c) = 3 sin (2π+ π/4)

(d) = 2 cos πt

Answer:

(a) https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m50ae5d.gif https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_4bbe19ab.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m7cdd4215.gif

If this equation is compared with the standard SHM equationhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m7df88465.gif , then we get:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m4b33c811.gif

The motion of the particle can be plotted as shown in the following figure.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m2c6c928.jpg

(b) https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m7057de71.gif https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_354abc50.gif

If this equation is compared with the standard SHM equationhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m7df88465.gif , then we get:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m173c140a.gif

The motion of the particle can be plotted as shown in the following figure.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m4d936fcc.jpg

(c) https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_504feafa.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m453a8b34.gif

If this equation is compared with the standard SHM equationhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m7df88465.gif , then we get:

Amplitude, A = 3 cm

Phase angle, https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_16c56f35.gif  = 135°

Angular velocity,https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m4ec0617.gif

The motion of the particle can be plotted as shown in the following figure.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_7398892a.jpg

(d) x = 2 cos πt

If this equation is compared with the standard SHM equationhttps://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m7df88465.gif , then we get:

Amplitude, A = 2 cm

Phase angle, Φ = 0

Angular velocity, ω = π rad/s

The motion of the particle can be plotted as shown in the following figure.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4702/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m5427f0f.jpg

Question 14.13:

Figure 14.30 (a) shows a spring of force constant clamped rigidly at one end and a mass attached to its free end. A force F applied at the free end stretches the spring. Figure 14.30 (b) shows the same spring with both ends free and attached to a mass at either end. Each end of the spring in Fig. 14.30(b) is stretched by the same force F.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m1cec2a12.jpg

(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?

Answer:

(a) For the one block system:

When a force F, is applied to the free end of the spring, an extension l, is produced. For the maximum extension, it can be written as:

F = kl

Where, k is the spring constant

Hence, the maximum extension produced in the spring, https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_me027210.gif

For the two block system:

The displacement (x) produced in this case is:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m4e473361.gif

Net force, F = +2 kx https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_796d3fc3.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_76b471d8.gif

(b) For the one block system:

For mass (m) of the block, force is written as:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m479b2a2b.gif

Where, x is the displacement of the block in time t

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_259baaa7.gif

It is negative because the direction of elastic force is opposite to the direction of displacement.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_1fc449d9.gif

Where,

ω is angular frequency of the oscillation

∴Time period of the oscillation, https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_1ab58cae.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_5797fa1e.gif

For the two block system:

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_56484d79.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_m10802a98.gif

It is negative because the direction of elastic force is opposite to the direction of displacement.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_187f20a6.gif

Where,

Angular frequency, https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_760dc93.gif

∴Time period, https://img-nm.mnimgs.com/img/study_content/curr/1/11/12/192/4703/NS_3-10-08_Sravana_11_Physis_14_25_NRJ_html_7568fc84.gif

Also Read : Page-No-361:-Chapter-14-Oscillations-class-11-ncert-solutions-Physics

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