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SIMPLE HARMONIC MOTION:--
OBJECTIVE-II
1. Select the correct statements.
(a) A simple harmonic motion is necessarily periodic.
(b) A simple harmonic motion is necessarily oscillatory.
(c) An oscillatory motion is necessarily periodic.
(d) A periodic motion is necessarily oscillatory.
ANSWER: (a), (b)
EXPLANATION: Since in an SHM the motion is to and fro (hence oscillatory), also it is repeated at a regular interval (hence periodic).
(c) is not true because all oscillatory motion is not repeated at regular interval. Example a person walking to and fro and thinking something but does not complete one full cycle in same time. His motion is oscillatory but not periodic. Another example is a ball falling on a hard floor which is not perfectly elastic. Each time its height and time taken to hit the ground varies. It is a vertical oscillation but not periodic.
(d) is not true because all periodic motions are not to and fro motion. For example a uniform circular motion.
2. A particle moves in a circular path with uniform speed. Its motion is
(a) periodic
(b) oscillatory
(c) simple harmonic
(d) angular simple harmonic.
ANSWER: (a)
EXPLANATION: Since a uniform circular motion repeats itself at a regular time interval.
(b) not true because it is not a to and fro motion.
(c) not true because SHM is a special type of oscillation.
(d) not true because the particle is not oscillating about a fixed axis.
3. A particle is fastened at the end of a string and is whirled in a vertical circle with the other end of the string being fixed. The motion of the particle is
(a) period
(b) oscillatory
(c) simple harmonic
(d) angular simple harmonic.
ANSWER: (a)
EXPLANATION: Same as in problem 2.
4. A particle moves in a circular path with a continuously increasing speed. Its motion is
(a) period
(b) oscillatory
(c) simple harmonic
(d) none of them.
ANSWER: (d)
EXPLANATION: Though the particle repeats its motion but not at a regular interval (Since the speed is continuously increasing), so (a) is not true.
(b) and (c) are not true because the motion is not a to and fro motion.
Since none of the (a), (b) and (c) is true hence (d) is true.
5. The motion of a torsional pendulum is
(a) period
(b) oscillatory
(c) simple harmonic
(d) angular simple harmonic.
ANSWER: (a), (b), (d)
EXPLANATION: (a) true because it repeats its motion at a regular time interval. (b) true because it is angular to and fro motion.
Since the oscillation is not on a straight line it is not a simple harmonic motion, (c) is not true.
(d) is true because the torque is directly proportional to the angular displacement and in opposite sense of it.
6. Which of the following quantities are always negative in a simple harmonic motion?
(a) F.a
(b) v.r
(c) a.r
(d) F.r.
ANSWER: (c), (d)
EXPLANATION: The dot product is the product of the magnitudes of the vectors and cosine of the angle between them. Only those options will be negative in which the vectors have always opposite directions i.e. the angle between them is π so that cosπ=-1.
F and a has always the same direction, so cos0=1, the option (a) is always positive hence not true.
v and r are sometimes in the same direction and sometimes in opposite direction, so their dot product is not always negative. Hence (b) is not true.
r and a are always in opposite directions hence their dot product will always be negative, hence (c) is true.
F and r are always in opposite directions hence their dot product will always be negative, hence (d) is true.
7. Which of the following quantities are always positive in a simple harmonic motion?
(a) F.a
(b) v.r
(c) a.r
(d) F.r.
ANSWER: (a)
EXPLANATION: Same as in problem 6.
F and a has always the same directions hence their dot product is always positive.
8. Which of the following quantities are always zero in a simple harmonic motion?
(a) Fxa
(b) vxr
(c) axr
(d) Fxr.
ANSWER: All.
EXPLANATION: The cross product is a vector itself and its magnitude is the product of the magnitudes of the vectors and the sine of the angle between them. If the vectors are parallel or anti-parallel i.e. if the angle between them is zero or π, its sine becomes zero and the cross product becomes zero.
Here all the vectors are along the same line either in the same direction or in opposite direction. Hence all options are zero.
9. Suppose a tunnel is dug along a diameter of the earth. A particle is dropped from a point, a distance h directly above the tunnel. The motion of the particle as seen from the earth is
(a) simple harmonic
(b) parabolic
(c) on a straight line
(d) periodic.
ANSWER: (c), (d)
EXPLANATION: Since the particle will cross the tunnel and go up to the height h on the opposite side and return back to the original point and repeat its same motion in the same time, so it is periodic. Since the tunnel is along the diameter the motion of the particle is on the straight line. Hence (c) and (d ) are true.
The force on the particle above the surface of the earth is inversely proportional to the square of its distance from the center of the earth, it is not directly proportional to that distance. Hence (a) is not true.
Tunnel along the diameter of the earth
(b) is not true because the motion is on a straight line.
10. For a particle executing simple harmonic motion, the acceleration is proportional to
(a) displacement from the mean position
(b) distance from the mean position
(c) distance traveled since t=0
(d) speed
ANSWER: (a)
EXPLANATION: Since a=-⍵²x, where x is the shortest instantaneous distance from the mean position i.e. magnitude of the displacement from the mean position. Hence (a) is true.
The distance in (b) is scaler and thus is not the shortest instantaneous distance from the mean position. So not true.
Due to the reason (b) and also at t=0 does not mean the mean position. (c) is not true.
Acceleration is not proportional to the speed in SHM, hence (d) not true.
11. A particle moves in the X-Y plane according to the equation
r = (i+2j)A.cos⍵t.
The motion of the particle is
(a) on a straight line
(b) on an ellipse
(c) periodic
(d) simple harmonic
ANSWER: (a), (c), (d)
EXPLANATION: r=A.cos⍵t is the equation of an SHM, i+2j is the direction of a straight line. So the given equation is the simple harmonic motion of the particle on a straight line, hence also periodic. So (a), (c) and (d) are true.
When the motion is on the straight line it can't be on an ellipse, so (b) is not true.
12. A particle moves on the X-axis according to the equation x=x₀sin²⍵t. The motion is simple harmonic
(a) with amplitude x₀
(b) with amplitude 2x₀
(c) with time period 2π/⍵
(d) with time period π/⍵
ANSWER: (d)
EXPLANATION: x = x₀sin²⍵t =x₀(1-cos2⍵t)/2
→x = x₀/2 - ½x₀.cos2⍵t.
It gives the amplitude as x₀/2, hence (a) and (b) are not true. The time period is given as T=2π/(2⍵) =π/⍵. So (d) is true not (c).
13. In a simple harmonic motion
(a) the potential energy is always equal to the kinetic energy
(b) the potential energy is never equal to the kinetic energy
(c) the average potential energy in any time interval is equal to the average kinetic energy in that time interval
(d) the average potential energy in one time period is equal to the average kinetic energy in this period.
ANSWER: (d)
EXPLANATION: The sum of K.E. and P.E. is constant, they are not equal. (a) is not true. Since their sum is constant at any instant, at some instant they may be equal. (b) is not true.
Near the mean position, K.E. is maximum while P.E. is minimum. Near the extremes, it is just the opposite. So if the time interval is chosen near the mean position or near the extremes, the average K.E. and P.E. will not be equal. (c) is not true.
The variations of the K.E. and P.E. are exactly similar in one time period except that K.E. is maximum at the mean position while P.E. is maximum at the extremes. So the average P.E. and K.E. are equal in one time period. (d) is true.
14. In a simple harmonic motion
(a) the maximum potential energy equals the maximum kinetic energy
(b) the minimum potential energy equals the minimum kinetic energy
(c) the minimum potential energy equals the maximum kinetic energy
(d) the maximum potential energy equals the minimum kinetic energy.
ANSWER: (a), (b)
EXPLANATION: Minimum P.E. and minimum K.E. are respectively at the mean position and at the extremes (equal to zero). And their maximums are just at the opposite points (equal to m⍵²A². So (a) and (b) are not true and (c), (d) are true.
15. An object is released from rest. The time it takes to fall through a distance h and the speed of the object as it falls through the distance is measured with a pendulum clock. The entire apparatus is taken on the moon and the experiment is repeated
(a) the measured times are same
(b) the measured speeds are same
(c) the actual times in the fall are equal
(d) the actual speeds are equal.
ANSWER: (a), (b)
EXPLANATION: The time of fall t = √(2h/g) and the final speed v=√(2gh). The time period of the pendulum clock T=2π√(l/g). If the acceleration due to the gravity at the moon is g' = g/6, the actual time of fall will be t' = √6.t, i.e. √6 time more than that on the earth. (c) is not true.
Similarly, the actual speed v' =v/√6, i.e. only 1/√6 th of v. (d) is not true.
Now the time period of the pendulum clock will be T'=√6.T, i.e. it will run √6 time slower than on the earth. Therefore the √6 times longer time of fall will be measured exactly the same by the √6 times slower clock. So (a) is true.
Since the clock shows the same time taken to fall through the same height, the measured speed will also be the same. (b) is true.
16. Which of the following will change the time period as they are taken to the moon?
(a) A simple pendulum
(b) A physical pendulum
(c) A torsional pendulum
(d) A spring mass-system.
ANSWER: (a), (b).
EXPLANATION: Since on the moon g changes, only that time period will change which depends on the value of g. The time period of a simple pendulum and the physical pendulum is a function of g so their time period will change on the moon. (a) and (b) are true.
The time period of a torsional pendulum and the spring-mass system is not dependent on the value of g, so they will not change on the moon. (c) and (d) not true.
1. Select the correct statements.
(a) A simple harmonic motion is necessarily periodic.
(b) A simple harmonic motion is necessarily oscillatory.
(c) An oscillatory motion is necessarily periodic.
(d) A periodic motion is necessarily oscillatory.
ANSWER: (a), (b)
EXPLANATION: Since in an SHM the motion is to and fro (hence oscillatory), also it is repeated at a regular interval (hence periodic).
(c) is not true because all oscillatory motion is not repeated at regular interval. Example a person walking to and fro and thinking something but does not complete one full cycle in same time. His motion is oscillatory but not periodic. Another example is a ball falling on a hard floor which is not perfectly elastic. Each time its height and time taken to hit the ground varies. It is a vertical oscillation but not periodic.
(d) is not true because all periodic motions are not to and fro motion. For example a uniform circular motion.
(c) is not true because all oscillatory motion is not repeated at regular interval. Example a person walking to and fro and thinking something but does not complete one full cycle in same time. His motion is oscillatory but not periodic. Another example is a ball falling on a hard floor which is not perfectly elastic. Each time its height and time taken to hit the ground varies. It is a vertical oscillation but not periodic.
(d) is not true because all periodic motions are not to and fro motion. For example a uniform circular motion.
2. A particle moves in a circular path with uniform speed. Its motion is
(a) periodic
(b) oscillatory
(c) simple harmonic
(d) angular simple harmonic.
ANSWER: (a)
EXPLANATION: Since a uniform circular motion repeats itself at a regular time interval.
(b) not true because it is not a to and fro motion.
(c) not true because SHM is a special type of oscillation.
(d) not true because the particle is not oscillating about a fixed axis.
(b) not true because it is not a to and fro motion.
(c) not true because SHM is a special type of oscillation.
(d) not true because the particle is not oscillating about a fixed axis.
3. A particle is fastened at the end of a string and is whirled in a vertical circle with the other end of the string being fixed. The motion of the particle is
(a) period
(b) oscillatory
(c) simple harmonic
(d) angular simple harmonic.
ANSWER: (a)
EXPLANATION: Same as in problem 2.
4. A particle moves in a circular path with a continuously increasing speed. Its motion is
(a) period
(b) oscillatory
(c) simple harmonic
(d) none of them.
ANSWER: (d)
EXPLANATION: Though the particle repeats its motion but not at a regular interval (Since the speed is continuously increasing), so (a) is not true.
(b) and (c) are not true because the motion is not a to and fro motion.
Since none of the (a), (b) and (c) is true hence (d) is true.
(b) and (c) are not true because the motion is not a to and fro motion.
Since none of the (a), (b) and (c) is true hence (d) is true.
5. The motion of a torsional pendulum is
(a) period
(b) oscillatory
(c) simple harmonic
(d) angular simple harmonic.
ANSWER: (a), (b), (d)
EXPLANATION: (a) true because it repeats its motion at a regular time interval. (b) true because it is angular to and fro motion.
Since the oscillation is not on a straight line it is not a simple harmonic motion, (c) is not true.
(d) is true because the torque is directly proportional to the angular displacement and in opposite sense of it.
Since the oscillation is not on a straight line it is not a simple harmonic motion, (c) is not true.
(d) is true because the torque is directly proportional to the angular displacement and in opposite sense of it.
6. Which of the following quantities are always negative in a simple harmonic motion?
(a) F.a
(b) v.r
(c) a.r
(d) F.r.
ANSWER: (c), (d)
EXPLANATION: The dot product is the product of the magnitudes of the vectors and cosine of the angle between them. Only those options will be negative in which the vectors have always opposite directions i.e. the angle between them is π so that cosπ=-1.
F and a has always the same direction, so cos0=1, the option (a) is always positive hence not true.v and r are sometimes in the same direction and sometimes in opposite direction, so their dot product is not always negative. Hence (b) is not true.
r and a are always in opposite directions hence their dot product will always be negative, hence (c) is true.
F and r are always in opposite directions hence their dot product will always be negative, hence (d) is true.
7. Which of the following quantities are always positive in a simple harmonic motion?
(a) F.a
(b) v.r
(c) a.r
(d) F.r.
ANSWER: (a)
EXPLANATION: Same as in problem 6.
F and a has always the same directions hence their dot product is always positive. 8. Which of the following quantities are always zero in a simple harmonic motion?
(a) Fxa
(b) vxr
(c) axr
(d) Fxr.
ANSWER: All.
EXPLANATION: The cross product is a vector itself and its magnitude is the product of the magnitudes of the vectors and the sine of the angle between them. If the vectors are parallel or anti-parallel i.e. if the angle between them is zero or π, its sine becomes zero and the cross product becomes zero.
Here all the vectors are along the same line either in the same direction or in opposite direction. Hence all options are zero.
Here all the vectors are along the same line either in the same direction or in opposite direction. Hence all options are zero.
9. Suppose a tunnel is dug along a diameter of the earth. A particle is dropped from a point, a distance h directly above the tunnel. The motion of the particle as seen from the earth is
(a) simple harmonic
(b) parabolic
(c) on a straight line
(d) periodic.
ANSWER: (c), (d)
EXPLANATION: Since the particle will cross the tunnel and go up to the height h on the opposite side and return back to the original point and repeat its same motion in the same time, so it is periodic. Since the tunnel is along the diameter the motion of the particle is on the straight line. Hence (c) and (d ) are true.
The force on the particle above the surface of the earth is inversely proportional to the square of its distance from the center of the earth, it is not directly proportional to that distance. Hence (a) is not true.
The force on the particle above the surface of the earth is inversely proportional to the square of its distance from the center of the earth, it is not directly proportional to that distance. Hence (a) is not true.
 |
| Tunnel along the diameter of the earth |
(b) is not true because the motion is on a straight line.
10. For a particle executing simple harmonic motion, the acceleration is proportional to
10. For a particle executing simple harmonic motion, the acceleration is proportional to
(a) displacement from the mean position
(b) distance from the mean position
(c) distance traveled since t=0
(d) speed
ANSWER: (a)
EXPLANATION: Since a=-⍵²x, where x is the shortest instantaneous distance from the mean position i.e. magnitude of the displacement from the mean position. Hence (a) is true.
The distance in (b) is scaler and thus is not the shortest instantaneous distance from the mean position. So not true.
Due to the reason (b) and also at t=0 does not mean the mean position. (c) is not true.
Acceleration is not proportional to the speed in SHM, hence (d) not true.
The distance in (b) is scaler and thus is not the shortest instantaneous distance from the mean position. So not true.
Due to the reason (b) and also at t=0 does not mean the mean position. (c) is not true.
Acceleration is not proportional to the speed in SHM, hence (d) not true.
11. A particle moves in the X-Y plane according to the equation
r = (i+2j)A.cos⍵t.
The motion of the particle is
(a) on a straight line
(b) on an ellipse
(c) periodic
(d) simple harmonic
ANSWER: (a), (c), (d)
EXPLANATION: r=A.cos⍵t is the equation of an SHM, i+2j is the direction of a straight line. So the given equation is the simple harmonic motion of the particle on a straight line, hence also periodic. So (a), (c) and (d) are true.
When the motion is on the straight line it can't be on an ellipse, so (b) is not true.
When the motion is on the straight line it can't be on an ellipse, so (b) is not true.
12. A particle moves on the X-axis according to the equation x=x₀sin²⍵t. The motion is simple harmonic
(a) with amplitude x₀
(b) with amplitude 2x₀
(c) with time period 2π/⍵
(d) with time period π/⍵
ANSWER: (d)
EXPLANATION: x = x₀sin²⍵t =x₀(1-cos2⍵t)/2
→x = x₀/2 - ½x₀.cos2⍵t.
It gives the amplitude as x₀/2, hence (a) and (b) are not true. The time period is given as T=2π/(2⍵) =π/⍵. So (d) is true not (c).
→x = x₀/2 - ½x₀.cos2⍵t.
It gives the amplitude as x₀/2, hence (a) and (b) are not true. The time period is given as T=2π/(2⍵) =π/⍵. So (d) is true not (c).
13. In a simple harmonic motion
(a) the potential energy is always equal to the kinetic energy
(b) the potential energy is never equal to the kinetic energy(c) the average potential energy in any time interval is equal to the average kinetic energy in that time interval
(d) the average potential energy in one time period is equal to the average kinetic energy in this period.
EXPLANATION: The sum of K.E. and P.E. is constant, they are not equal. (a) is not true. Since their sum is constant at any instant, at some instant they may be equal. (b) is not true.
Near the mean position, K.E. is maximum while P.E. is minimum. Near the extremes, it is just the opposite. So if the time interval is chosen near the mean position or near the extremes, the average K.E. and P.E. will not be equal. (c) is not true.
The variations of the K.E. and P.E. are exactly similar in one time period except that K.E. is maximum at the mean position while P.E. is maximum at the extremes. So the average P.E. and K.E. are equal in one time period. (d) is true.
Near the mean position, K.E. is maximum while P.E. is minimum. Near the extremes, it is just the opposite. So if the time interval is chosen near the mean position or near the extremes, the average K.E. and P.E. will not be equal. (c) is not true.
The variations of the K.E. and P.E. are exactly similar in one time period except that K.E. is maximum at the mean position while P.E. is maximum at the extremes. So the average P.E. and K.E. are equal in one time period. (d) is true.
14. In a simple harmonic motion
(a) the maximum potential energy equals the maximum kinetic energy
(b) the minimum potential energy equals the minimum kinetic energy
(c) the minimum potential energy equals the maximum kinetic energy
(d) the maximum potential energy equals the minimum kinetic energy.
ANSWER: (a), (b)
EXPLANATION: Minimum P.E. and minimum K.E. are respectively at the mean position and at the extremes (equal to zero). And their maximums are just at the opposite points (equal to m⍵²A². So (a) and (b) are not true and (c), (d) are true.
15. An object is released from rest. The time it takes to fall through a distance h and the speed of the object as it falls through the distance is measured with a pendulum clock. The entire apparatus is taken on the moon and the experiment is repeated
(a) the measured times are same
(b) the measured speeds are same
(c) the actual times in the fall are equal
(d) the actual speeds are equal.
EXPLANATION: The time of fall t = √(2h/g) and the final speed v=√(2gh). The time period of the pendulum clock T=2π√(l/g). If the acceleration due to the gravity at the moon is g' = g/6, the actual time of fall will be t' = √6.t, i.e. √6 time more than that on the earth. (c) is not true.
Similarly, the actual speed v' =v/√6, i.e. only 1/√6 th of v. (d) is not true.
Now the time period of the pendulum clock will be T'=√6.T, i.e. it will run √6 time slower than on the earth. Therefore the √6 times longer time of fall will be measured exactly the same by the √6 times slower clock. So (a) is true.
Since the clock shows the same time taken to fall through the same height, the measured speed will also be the same. (b) is true.
16. Which of the following will change the time period as they are taken to the moon?
(a) A simple pendulum
(b) A physical pendulum
(c) A torsional pendulum
(d) A spring mass-system.
EXPLANATION: Since on the moon g changes, only that time period will change which depends on the value of g. The time period of a simple pendulum and the physical pendulum is a function of g so their time period will change on the moon. (a) and (b) are true.
The time period of a torsional pendulum and the spring-mass system is not dependent on the value of g, so they will not change on the moon. (c) and (d) not true.
(b) the measured speeds are same
(c) the actual times in the fall are equal
(d) the actual speeds are equal.
ANSWER: (a), (b)
EXPLANATION: The time of fall t = √(2h/g) and the final speed v=√(2gh). The time period of the pendulum clock T=2π√(l/g). If the acceleration due to the gravity at the moon is g' = g/6, the actual time of fall will be t' = √6.t, i.e. √6 time more than that on the earth. (c) is not true.
Similarly, the actual speed v' =v/√6, i.e. only 1/√6 th of v. (d) is not true.
Now the time period of the pendulum clock will be T'=√6.T, i.e. it will run √6 time slower than on the earth. Therefore the √6 times longer time of fall will be measured exactly the same by the √6 times slower clock. So (a) is true.
Since the clock shows the same time taken to fall through the same height, the measured speed will also be the same. (b) is true.
16. Which of the following will change the time period as they are taken to the moon?
(a) A simple pendulum
(b) A physical pendulum
(c) A torsional pendulum
(d) A spring mass-system.
ANSWER: (a), (b).
The time period of a torsional pendulum and the spring-mass system is not dependent on the value of g, so they will not change on the moon. (c) and (d) not true.
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Links to the chapters -
CHAPTER- 11 - Gravitation
EXERCISES -Q 31 TO 39
CHAPTER- 10 - Rotational Mechanics
CHAPTER- 9 - Center of Mass, Linear Momentum, Collision
Links to the chapters -
CHAPTER- 11 - Gravitation
CHAPTER- 10 - Rotational Mechanics
CHAPTER- 9 - Center of Mass, Linear Momentum, Collision
HC Verma's Concepts of Physics, Chapter-8, WORK AND ENERGY
Click here for → Question for Short Answers
Click here for → OBJECTIVE-I
Click here for → OBJECTIVE-II
Click here for → Exercises (1-10)
Click here for → Question for Short Answers
Click here for → OBJECTIVE-I
Click here for → OBJECTIVE-II
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Click here for → Exercises (11-20)
Click here for → Exercises (21-30)
Click here for → Exercises (31-42)
Click here for → Exercise(43-54)
Click here for → Exercises (21-30)
Click here for → Exercises (31-42)
Click here for → Exercise(43-54)
Click here for → Exercises (31-42)
Click here for → Exercise(43-54)
HC Verma's Concepts of Physics, Chapter-7, Circular Motion
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Click here for → OBJECTIVE-I
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HC Verma's Concepts of Physics, Chapter-6, Friction
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Click here for → OBJECTIVE-I
Click here for → Friction - OBJECTIVE-II
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For more practice on problems on friction solve these "New Questions on Friction" .
Click here for → Friction - OBJECTIVE-II
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For more practice on problems on friction solve these "New Questions on Friction" .
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HC Verma's Concepts of Physics, Chapter-5, Newton's Law's of Motion
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Click here for→Newton's Laws of Motion,Exercises(Q.No. 13 to 27)
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HC Verma's Concepts of Physics, Chapter-4, The Forces
"Questions for short Answers"
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Click here for "The Forces" - Exercises
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very nice explanation sir.please provide solutions to rest of chapters too.
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