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A body of mass m is attached by an inela...

A body of mass `m` is attached by an inelastic string to a suspended spring of spring constant `k`. Both the string and the spring have negligible mass and the string is inextensible and of length `L`. Initially, the mass `m` is at rest.
The largest amplitude `A_(max)`, for which the string will remain taut throughout the motion is

A

`(mg)/(2k)`

B

`(mg)/(k)`

C

`L//2`

D

`L`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to determine the largest amplitude \( A_{\text{max}} \) for which the inelastic string remains taut during the oscillation of the mass \( m \) attached to the spring with spring constant \( k \). ### Step-by-Step Solution: 1. **Understanding the System**: - We have a mass \( m \) attached to a spring with spring constant \( k \) via an inelastic string of length \( L \). - The mass is initially at rest, and we need to analyze its motion when it oscillates. 2. **Condition for the String to Remain Taut**: - For the string to remain taut, the tension in the string must not become zero at any point during the oscillation. - This means that when the mass is at its maximum displacement (amplitude), the spring must be stretched such that the force exerted by the spring equals the weight of the mass. 3. **Applying Hooke's Law**: - According to Hooke's law, the force exerted by the spring when it is stretched by a distance \( x \) is given by: \[ F_{\text{spring}} = kx \] - At maximum amplitude \( A_{\text{max}} \), the force exerted by the spring must balance the weight of the mass: \[ kx = mg \] 4. **Finding the Maximum Displacement**: - From the equation \( kx = mg \), we can solve for \( x \): \[ x = \frac{mg}{k} \] - This \( x \) represents the maximum stretch of the spring when the mass is at its lowest point. 5. **Conclusion**: - Since the maximum amplitude \( A_{\text{max}} \) must be equal to the maximum stretch \( x \) for the string to remain taut, we have: \[ A_{\text{max}} = \frac{mg}{k} \] ### Final Answer: The largest amplitude \( A_{\text{max}} \) for which the string will remain taut throughout the motion is: \[ A_{\text{max}} = \frac{mg}{k} \]
To solve the problem, we need to determine the largest amplitude \( A_{\text{max}} \) for which the inelastic string remains taut during the oscillation of the mass \( m \) attached to the spring with spring constant \( k \). ### Step-by-Step Solution: 1. **Understanding the System**: - We have a mass \( m \) attached to a spring with spring constant \( k \) via an inelastic string of length \( L \). - The mass is initially at rest, and we need to analyze its motion when it oscillates. ...
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