Video Lecture
Theory For Making Notes
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Practice Questions (Basic Level)
Q.1
What the equation of SHM with amplitude 8 cm and maximum particle speed 64p cm/s (where y is also in cm.)
(a) y = 64 sin 2 pt
(b) y = 8 sin 8 pt
(c) y = 64 sin 8 pt
(d) y = 8 sin 64 pt
Ans : (b)
Q.2
A particle moves such that its acceleration is given by a = -b2 What is the period of oscillation
(a) 2p/b (b) 2p/8b (c) b/2p (d) 2p/3b
Ans : (a)
Q.3
The displacement x (in centimeters) of an oscillating particle varies with time t (in seconds) as \displaystyle x=2\cos \left( {0.5\ \pi t+\frac{\pi }{3}} \right)The magnitude of the maximum acceleration of the particle in cms–2 is
(a) \displaystyle \frac{\pi }{2}
(b) \displaystyle \frac{\pi }{4}
(c) \displaystyle \frac{{{{\pi }^{2}}}}{2}
(d) \displaystyle \frac{{{{\pi }^{2}}}}{4}
Ans : (c)
Q.4
Two SHM’s are respectively represented by y = a sin(wt – kx) and y = b cos(wt – kx). Then the phase difference between the two is
(a) p/2 (b) p/4 (c) p/3 (d) None
Ans : (a)
Q.5
A particle executes SHM. Its velocities are v1 and v2 at displacement x1 and x2 from mean position respectively. The frequency of oscillation will be:
(a) \displaystyle 1000\sqrt{3}
(b) \displaystyle \frac{{2\sqrt{\beta }}}{{1+\beta }}
(c) \displaystyle \begin{array}{l}\frac{2}{{\left( {1+\beta } \right)}}+\\\end{array}
(d) \displaystyle \frac{{\sqrt{\beta }}}{{1+\beta }}
Ans. (b)
Q.6
Which of the following is a simple harmonic motion?
(a) Ball bouncing between two rigid vertical walls
(b) Particle moving in a circle with uniform speed
(c) Wave moving through a string fixed at both ends
(d) Earth spinning about its own axis
Ans. (c)
Q.7
Which one of the following statements is true for the speed v and the acceleration a of a particle executing simple harmonic motion?
(a) When v is maximum, a is maximum
(b) Value of a is zero, whatever may be the value of v
(c) When v is zero, a is zero
(d) When v is maximum, a is zero
Ans. (d)
Q.8
A particle executing simple harmonic motion of amplitude 5 cm has maximum speed of 31.4cm/s. The frequency of its oscillation is
(a) 3 Hz (b) 2 Hz (c) 4 Hz (d) 1 Hz
Ans. (d)
Q.9
The phase (at a time t) of a particle in simple harmonic motion tells:
(a) Only the position of the particle at time t
(b) One the direction of motion of the particle at time t
(c) Both the position and direction of motion of the particle at time t
(d) Neither the position of the particle nor its direction of motion at time t
Ans. (c)
Q.10
The displacement of a particle executing SHM is given by y = 0.25 sin 200 t cm. The maximum speed of the particle is
(a) 200 cm \displaystyle {{\in }_{0}}\,\,L\left( {\frac{{\Delta V}}{{\Delta t}}} \right),
(b) 100 cm \displaystyle \rho =KL+\frac{{MI}}{\omega },
(c) 50 cm \frac{{\Delta v}}{{\Delta z}}
(d) 5.25 cm \eta A\frac{{dv}}{{dz}},
Ans. (a)
Q.11
The displacement of a particle executing SHM is given by x = 0.01 sin 100 \displaystyle \eta (t + 0.05). The time period is
(a) 0.01 s (b) 0.02 s (c) 0.1 s (d) 0.2 s
Ans. (b)
Q.12
A body is vibrating in simple harmonic motion. If its acceleration is 12 cm/s2 at a displacement 3 cm from the mean position, then time period is
(a) 6.28 s (b) 3.14 s (c) 1.57 s (d) 2.57
Ans. (b)
Q.13
The average acceleration of a particle performing SHM over one complete oscillation is
(a) \frac{\pi }{2}
(b) {{\cos }^{{-1}}}(0.6)
(c) zero
(d) {{\tan }^{{-1}}}\left( {\frac{7}{5}} \right)
Ans. (c)
Practice Questions (JEE Main Level)
Q.1
A rough horizontal table moves horizontally in SHM, the period being 3 s and the maximum speed 4 m/s. A small heavy mass is placed on the table. Then the least coefficient of friction if the mass does not slide on the table throughout the motion.
(a) m = 0.85 (b) m = 0.65 (c) m = 0.55 (d) m = 0.25
Ans : (a)
Q.2
In how much time a particle will travel from its mean position to a displacement equal to half of its amplitude undergoing SHM with a time period of 2 seconds.
(a) 2/7s
(b) \displaystyle \frac{1}{6}s
(c) 1/2s
(d) 2/3s
Ans : (b)
Q.3
A person normally weighing 60 kg stands on a platform which oscillates up and down harmonically at a frequency 2.0 sec–1 and an amplitude 5.0 cm. If a machine on the platform gives the person’s weight against time, deduce the maximum and minimum reading it will show, take g = 10 m/sec2.
(a) Maximum reading = 20.3 kg, Minimum reading = 14.7 kg.
(b) Maximum reading = 107.3 kg, Minimum reading = 102.7 kg.
(c) Maximum reading = 107.3 kg, Minimum reading = 12.7 kg.
(d) None
Ans : (c)
Q.4
Two SHMs are given by y1 = a sin [(p/2)t + f] and \displaystyle \frac{3}{2}P The phase difference between these after 1 sec is:
(a) p (b) p/2 (c) p/4 (d) p/6
Ans. (d)
Q.5
A body of mass 5 g is executing SHM about a point O with amplitude 100 cm. If its max. velocity is 100 cm/sec, its velocity will be 50 cm/sec at a distance (in cm):
(a) 5
(b) \displaystyle \frac{{f\mu \prime \left( {\mu 1} \right)}}{{\left( {\mu \mu \prime } \right)}}
(c) \displaystyle \frac{{f\left( {\mu \prime \mu } \right)}}{{\mu \prime \left( {\mu 1} \right)}}
(d) \displaystyle \frac{{\mu \prime \left( {\mu 1} \right)}}{{f\left( {\mu \prime \mu } \right)}}
Ans. (c)
Q.6
A horizontal platform is made to execute SHM of amplitude a in the vertical direction. An object placed on the platform will lose contact with it, when the frequency of oscillation exceeds:
(a) \displaystyle {{\sin }^{{1}}}\left( {\frac{8}{9}} \right)
(b) \displaystyle {{I}_{0}}\cos \left( {\frac{x}{\beta }} \right)
(c) \displaystyle {{I}_{0}}{{\cos }^{2}}\left( {\frac{x}{\beta }} \right)
(d) \displaystyle {{I}_{0}}{{\cos }^{2}}\left( {\frac{{\pi x}}{\beta }} \right)
Ans. (b)
Q.7
A particle performs SHM along a straight line with the period T and amplitude A. The mean velocity of the particle averaged over the time interval during which it travels a distance A/2 starting from the extreme position is:
(a) \displaystyle \left( {\frac{{{{I}_{0}}}}{4}} \right){{\cos }^{2}}\left( {\frac{{\pi x}}{\beta }} \right)
(b) \displaystyle \frac{\lambda }{{4(\mu -1)}}
(c) \displaystyle \frac{\lambda }{{2(\mu -1)}}
(d) \displaystyle \frac{\lambda }{{\mu -1}}
Ans. (d)
Q.8
The time period and the amplitude pendulum are 4 second and 0.20 metre respectively. If the displacement is 0.1 m at time t = 0, the equation of its displacement is represented by:
(a) y = 0.2 sin (0.5pt)
(b) y = 0.2 sin (0.5pt)
(c) y = 0.1 sin (pt + p/6)
(d) y = 0.1 sin (0.5pt)
Ans. (b)
Q.9
The function sin2 (wt) represents:
(a) A periodic, but not simple harmonic motion with a period \displaystyle \sqrt{{\frac{{3\lambda D}}{2}}}
(b) A periodic, but not simple harmonic motion with a period \displaystyle \sqrt{{\lambda D}}
(c) A simple harmonic motion with a period \displaystyle \sqrt{{\frac{{\lambda D}}{2}}}
(d) A simple harmonic motion with a period \displaystyle \sqrt{{3\lambda D}}
Ans. (b)
Q.10
A particle executes linear simple harmonic motion with an amplitude of 2 cm. When the particle is at 1 cm from the mean position the magnitude of its velocity is equal to that of its acceleration. Then its time period in second is:
(a) \displaystyle {{\cos }^{{1}}}\sqrt{{\frac{{12}}{{50}}}}
(b) \displaystyle \frac{1}{{\sqrt{2}}}m/{{s}^{2}}\,\,towards\,\,northeast
(c) \displaystyle \frac{1}{{\sqrt{2}}}m/{{s}^{2}}\,\,towards\,\,northwest
(d) \displaystyle 1\text{/}2\,\,m/{{s}^{2}}\,\,towards\,\,northwest
Ans. (c)
Q.11
The speed (v) of a particle moving along a straight line, when it is at a distance (x) from a fixed point on the line, is given v2 = 144 – 9x2. Select wrong alternate:
(a) Displacement of the particle £ distance moved by it
(b) The magnitude of acceleration at a distance 3 units from the fixed point is 27 units
(c) The motion is simple harmonic with \displaystyle \frac{{{{E}^{2}}}}{{{{\mu }_{0}}}} units
(d) The maximum displacement from the fixed point is 4 units
Ans. (c)
Q.12
A body executes SHM whose period is 16 s. Two seconds after it passes the equilibrium position, its velocity is 1 ms-1. The amplitude of SHM is:
(a) 6.3 m (b) 1.8 m (c) 3.6 m (d) 2.4 m
Ans. (c)
Q.13
For a particle in SHM, if the amplitude of the displacement is a and the amplitude of velocity is v, the amplitude of acceleration is
(a) va
(b) \frac{\pi }{4}
(c) \overset{\to }{\mathop{A}}\,\,\,and\,\,\overset{\to }{\mathop{B}}\,
(d) \overset{\to }{\mathop{A}}\,\cdot (\overset{\to }{\mathop{B}}\,\times \overset{\to }{\mathop{A}}\,)
Ans. (d)
Q.14
Out of the following functions representing motion of a particle which represents SHM
- \overset{\to }{\mathop{A}}\,
- \overset{\to }{\mathop{B}}\,
- |\overset{\to }{\mathop{A}}\,+\overset{\to }{\mathop{B}}\,|=|\overset{\to }{\mathop{A}}\,-\overset{\to }{\mathop{B}}\,|
- \overset{\to }{\mathop{A}}\,=3\hat{i}+4\hat{j}+5\hat{k}
(a) only IV does not represent SHM
(b) I and III
(c) I and II
(d) only I
Ans. (b)
Q.15
The displacement time graph of a particle executing SHM is shown in figure. Which of the following statement is false?
(a) The acceleration is maximum at t = T
(b) The force is zero at \displaystyle \frac{\pi }{{30\sqrt{2}}}cm/sa
(c) The potential energy equals the total oscillation energy \displaystyle \vec{P},\,\,\vec{Q},\,\,and\,\,R
(d) None of the above
Ans. (d)
Practice Questions (JEE Advance Level)
1.
A body oscillates with SHM, according to the equation
x=\left( {5.0} \right)m cos [(2p rad/s) t + p/4]
At t = 1.5 s, calculate
(a)
Displacement
(a)-2.765 m
(b)2.765 m
(c)3.535 m
(d)-3.535 m
Ans (d)
(b)
speed
(a) 2.48 m/s
(b)18.58 m/s
(c)20.67 m/s
(d)22.22 m/s
Ans (d)
(c)
acceleration of the body.
(a)130.35 m/s2
(b)132.62 m/s2
(c)139.56 m/s2
(d)150.45 m/s2
Ans (c)
2.
The displacement from equilibrium of a particle is given by x = Acos(w t – p/3). Which, if any, of the following are equivalent expressions
(a) x = Acos(w t + p/3)
(b) x = Asin(wt + p/6)
(c) x = A cos(wt + 5p/3)
(d) x = A sin(wt – 5p/6)
Ans (b) and (c)
3.
Which of the following functions are
(a) aperiodic (b) periodic but not simple harmonic (c) simple harmonic
(i) sin 2wt (ii) 1 + cos 2wt (iii) a sin wt + b cos wt (iv) sin wt + sin 2wt + cos 3wt
(v) sin3wt (vi) log (1 + wt) (vii) exp (-wt)
Ans (a) Functions (vi), (vii) are a periodic.
(b) Functions (ii), (iv) and (v) are periodic but not simple harmonic.
(c) Functions (i) and (iii) are simple harmonic.
4.
The condition |v| = 0.5 vmax, where vmaxis the maximum speed, occurs four times in each cycle in the oscillation of a block-spring system. Determine the first four time (>0) given that the displacement from equilibrium is x = 0.35 cos(3.6t – 0.5) m.
(a)0.284 s, 0.866 s, 1.16 s, 0.4 s
(b)0.284 s, 0.866 s, 1.16 s, 8.32 s
(c)0.284 s, 0.866 s, 2.3 s, 0.3 s
(d)0.284 s, 0.866 s, 1.16 s, 1.74 s
Ans (d)
5.
A particle of mass m is moving in a force field whose potential energy is given by u = a + bx2 J/kg. Show that the motion of particle is simple harmonic. Find the frequency in Hz.
(a) \displaystyle \left( {\frac{3}{\pi }\,\,\sqrt{{\frac{a}{5}}}} \right)
(b) \displaystyle \left( {\frac{1}{\pi }\,\,\sqrt{{\frac{a}{2}}}} \right)
(c) \displaystyle \left( {\frac{1}{\pi }\,\,\sqrt{{\frac{a}{3}}}} \right)
(d) \displaystyle \left( {\frac{1}{\pi }\,\,\sqrt{{\frac{a}{2}}}} \right)
Ans (d)
6.
The average speed over the period of a complete oscillation of a particle performing rectilinear SHM is 2 cm/s. If the particle attains this speed when it is at a point P whose distance from the centre O is 4 cm, determine the amplitude
(a)1.125 cm
(b)2.154 cm
(c)0.56 cm
(d)1.075 cm
Ans (a)
7.
A particle in SHM is described by the displacement function
x=A\cos \left( {\omega t+\varphi } \right), w = 2p/T
If the initial (t = 0) position of the particle is 1 cm and its initial velocity is p cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is p s-1.
(a)2p/4
(b)p/4
(c)3p/4
(d)8p/5
Ans (c)
8.
A spring balance has a scale that reads 50 kg. The length of the scale is 20 cm. A body suspended from this spring, when displaced and released, oscillates with period of 0.60 s. What is the weight of the body?
(a)219.1 N = 21.04 kgf
(b)219.1 N = 20.03 kgf
(c)219.1 N = 22.36 kgf
(d)219.1 N = 25.55 kgf
Ans (c)