A 10 kg stone is suspended with a rope of breaking strength 30 kg-wt. The minimum time in which the stone can be raised through a height 10 m starting from rest is (Take, `g = 10 N kg^(-1)`).

Options

A. 0.5s
B. 1.0s
C. `sqrt((2)/( 3)) s `
D. 2.0s

Answer

To solve the problem, we need to determine the minimum time required to raise a 10 kg stone through a height of 10 meters, given that the rope has a breaking strength of 30 kg-wt. We will follow these steps:

### Step 1: Calculate the Maximum Force
The breaking strength of the rope is given as 30 kg-wt. We can convert this to Newtons using the acceleration due to gravity (g = 10 N/kg).

\[ \text{Breaking Strength} = 30 \, \text{kg-wt} = 30 \times g = 30 \times 10 \, \text{N} = 300 \, \text{N} \]

### Step 2: Calculate the Maximum Acceleration
The maximum force that can be applied is 300 N. The mass of the stone is 10 kg. We can use Newton's second law to find the maximum acceleration (a_max) that can be applied to the stone.

\[ F = m \cdot a \implies a_{\text{max}} = \frac{F}{m} = \frac{300 \, \text{N}}{10 \, \text{kg}} = 30 \, \text{m/s}^2 \]

### Step 3: Account for Gravitational Acceleration
The stone is also subject to the acceleration due to gravity, which acts downward. Therefore, the effective acceleration (a_eff) with which we can raise the stone is given by:

\[ a_{\text{eff}} = a_{\text{max}} - g = 30 \, \text{m/s}^2 - 10 \, \text{m/s}^2 = 20 \, \text{m/s}^2 \]

### Step 4: Use the Equation of Motion
We will use the equation of motion to find the time (t) it takes to raise the stone through a height (s) of 10 meters. The equation is:

\[ s = ut + \frac{1}{2} a t^2 \]

Since the stone starts from rest, the initial velocity (u) is 0. Thus, the equation simplifies to:

\[ s = \frac{1}{2} a_{\text{eff}} t^2 \]

Substituting the values we have:

\[ 10 \, \text{m} = \frac{1}{2} \cdot 20 \, \text{m/s}^2 \cdot t^2 \]

### Step 5: Solve for Time
Now we can solve for t:

\[ 10 = 10 t^2 \implies t^2 = 1 \implies t = \sqrt{1} = 1 \, \text{s} \]

### Final Answer
The minimum time in which the stone can be raised through a height of 10 meters is **1 second**.

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