Home
Class 11
PHYSICS
An object of mass 4 kg is attached to a ...

An object of mass 4 kg is attached to a spring having spring constant `100(N)/(m)`. It performs simple harmonic motion on a smooth horizontal surface with an amplitude of 2 m. A 6 kg object is dropped vertically onto the 4 kg object when it crosses the mean position, and sticks to it. the change in amplitude of oscillation due to collision is

A

`1m`

B

`zero`

C

`2[1-sqrt((2)/(5))]`

D

`2[1-(1)/(sqrt5)]`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the change in amplitude of the oscillation after a 6 kg object is dropped onto a 4 kg object that is attached to a spring and is performing simple harmonic motion. ### Step-by-step Solution: 1. **Identify Initial Parameters:** - Mass of the first object (m1) = 4 kg - Spring constant (k) = 100 N/m - Amplitude (A) = 2 m 2. **Calculate Initial Velocity (v0) at Mean Position:** - At the mean position, the velocity is maximum. The total mechanical energy in simple harmonic motion is given by: \[ E = \frac{1}{2} k A^2 \] - Substitute the values: \[ E = \frac{1}{2} \times 100 \times (2)^2 = \frac{1}{2} \times 100 \times 4 = 200 \text{ J} \] - The kinetic energy at the mean position is equal to the total energy: \[ \frac{1}{2} m_1 v_0^2 = 200 \] - Rearranging gives: \[ v_0^2 = \frac{400}{4} = 100 \implies v_0 = 10 \text{ m/s} \] 3. **Apply Conservation of Momentum:** - When the 6 kg object is dropped onto the 4 kg object, momentum is conserved. - Initial momentum before collision: \[ p_{initial} = m_1 v_0 = 4 \times 10 = 40 \text{ kg m/s} \] - Let the final velocity after the collision be \( v' \): \[ p_{final} = (m_1 + m_2) v' = (4 + 6) v' = 10 v' \] - Setting initial momentum equal to final momentum: \[ 40 = 10 v' \implies v' = 4 \text{ m/s} \] 4. **Calculate New Amplitude (A'):** - The total mechanical energy after the collision is given by: \[ E' = \frac{1}{2} (m_1 + m_2) (v')^2 = \frac{1}{2} \times 10 \times (4)^2 = \frac{1}{2} \times 10 \times 16 = 80 \text{ J} \] - The new amplitude \( A' \) can be found using the energy conservation equation: \[ E' = \frac{1}{2} k (A')^2 \] - Substitute the values: \[ 80 = \frac{1}{2} \times 100 \times (A')^2 \] - Rearranging gives: \[ 80 = 50 (A')^2 \implies (A')^2 = \frac{80}{50} = \frac{8}{5} \implies A' = \sqrt{\frac{8}{5}} = \frac{2\sqrt{2}}{\sqrt{5}} \text{ m} \] 5. **Calculate Change in Amplitude:** - The change in amplitude \( \Delta A \) is: \[ \Delta A = A' - A = \frac{2\sqrt{2}}{\sqrt{5}} - 2 \] ### Final Result: The change in amplitude of oscillation due to the collision is: \[ \Delta A = \frac{2\sqrt{2}}{\sqrt{5}} - 2 \]

To solve the problem, we need to find the change in amplitude of the oscillation after a 6 kg object is dropped onto a 4 kg object that is attached to a spring and is performing simple harmonic motion. ### Step-by-step Solution: 1. **Identify Initial Parameters:** - Mass of the first object (m1) = 4 kg - Spring constant (k) = 100 N/m - Amplitude (A) = 2 m ...
Promotional Banner

Topper's Solved these Questions

  • LINEAR AND ANGULAR SIMPLE HARMONIC MOTION

    CENGAGE PHYSICS ENGLISH|Exercise Multiple Correct|35 Videos
  • LINEAR AND ANGULAR SIMPLE HARMONIC MOTION

    CENGAGE PHYSICS ENGLISH|Exercise Assertion Reasoning|6 Videos
  • LINEAR AND ANGULAR SIMPLE HARMONIC MOTION

    CENGAGE PHYSICS ENGLISH|Exercise Subjective|21 Videos
  • KINETIC THEORY OF GASES AND FIRST LAW OF THERMODYNAMICS

    CENGAGE PHYSICS ENGLISH|Exercise Interger|11 Videos
  • MISCELLANEOUS KINEMATICS

    CENGAGE PHYSICS ENGLISH|Exercise Interger type|3 Videos

Similar Questions

Explore conceptually related problems

A particle of mass 3 kg , attached to a spring with force constant 48Nm^(-1) execute simple harmonic motion on a frictionless horizontal surface. The time period of oscillation of the particle, is seconds , is

a block of mass 0.1 kg attached to a spring of spring constant 400 N/m pulled horizontally from x=0 to x_1 =10 mm. Find the work done by the spring force

An object of mass m is performing simple harmonic motion on a smooth horizontal surface as shown in Fig. just as the oscillating object reaches its extreme position, another object of mass 2 m is dropped on to oscillating object, which sticks to it. For this situation mark out the correct statement (s).

An object of mass M and radius R is performing pure rolling motion on a smooth horizontal surface under the action of a constant force F as shown in Fig. The object may be

A block with mass M attached to a horizontal spring with force constant k is moving with simple harmonic motion having amplitude A_(1) . At the instant when the block passes through its equilibrium position a lump of putty with mass m is dropped vertically on the block from a very small height and sticks to it. (a) Find the new amplitude and period. (b) Repeat part (a) for the case in which the putty is dropped on the block when it is at one end of its path.

A block with mass M attached to a horizontal spring with force constant k is moving with simple harmonic motion having amplitude A_(1) . At the instant when the block passes through its equilibrium position a lump of putty with mass m is dropped vertically on the block from a very small height and sticks to it. (a) Find the new amplitude and period. (b) Repeat part (a) for the case in which the putty is dropped on the block when it is at one end of its path.

Two blocks of masses 6 kg and 3 kg are attached to the two ends of a massless spring of spring constant 2 pi^(2) N//m . If spring is compressed and released on a smooth horizontal surface then find the time period (in seconds) of each block.

A mass M is attached to a horizontal spring of force constant k fixed one side to a rigid support as shown in figure. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass M is in equilibrium position, another mass m is gently placed on it. When will be the new amplitude of ocillation?

Passage III A block of mass m =1/2 kg is attached with two springs each of force contant k=10N/m as shown. The block is oscillating vertically up and down with amplitude A=25cm. When the block is at its mean position one of the spring breaks without changing momentum of block. What is the new amplitude of osciallation of the block.

A 2kg mass is attached to a spring of force constant 600 N//m and rests on a smooth horizontal surface. A second mass of 1kg slides along the surface toward the first at 6m//s . (a) Find the amplitude of oscillation if the masses make a perfectly inelastic collision and remain together on the spring. what is the period of oscillation ? (b) Find the amplitude and period of oscillation if the collision is perfectly elastic. ( c) For each case , write down the position x as a function of time t for the mass attached to the spring, assuming that the collision occurs at time t = 0 . What is the impulse given to the 2kg mass in each case ?

CENGAGE PHYSICS ENGLISH-LINEAR AND ANGULAR SIMPLE HARMONIC MOTION-Single Correct
  1. A particle performs simple harmonic motion about O with amolitude A an...

    Text Solution

    |

  2. In the previous question, the magnitude of velocity of particle at the...

    Text Solution

    |

  3. An object of mass 4 kg is attached to a spring having spring constant ...

    Text Solution

    |

  4. A cork floating on the pond water executes a simple harmonic motion, m...

    Text Solution

    |

  5. A spring is placed in vertical position by suspending it from a hook a...

    Text Solution

    |

  6. A particle is performing SHM according to the equation x=(3cm)sin((2pi...

    Text Solution

    |

  7. Which one is not correct for a cyclic process as shown in the figure ?

    Text Solution

    |

  8. A particle of mass m is present in a region where the potential energy...

    Text Solution

    |

  9. A particle performing simple harmonic motion having time period 3 s is...

    Text Solution

    |

  10. A particle performs SHM on x- axis with amplitude A and time period ...

    Text Solution

    |

  11. The coefficient of friction between block of mass m and 2m is mu=2tant...

    Text Solution

    |

  12. A block of mass m is suspended from a spring and executes vertical SHM...

    Text Solution

    |

  13. A particle performs SHM of amplitude A along a straight line. When it ...

    Text Solution

    |

  14. A horizontal spring -block system of mass 2kg executes S.H.M when the ...

    Text Solution

    |

  15. A metre stick swinging in vertical plane about a fixed horizontal axis...

    Text Solution

    |

  16. A physical pendulum is positioned so that its centre of gravity is abo...

    Text Solution

    |

  17. A particle executing harmonic motion is having velocities v1 and v2 at...

    Text Solution

    |

  18. A wire is bent an an angle theta. A rod of mass m can slide along the ...

    Text Solution

    |

  19. A solid right circular cylinder of weight 10 kg and cross sectional ar...

    Text Solution

    |

  20. A block A of mass m is placed on a smooth horizontal platform P and be...

    Text Solution

    |