A mass of 1 kg is suspended by means of a thread. The system is (i) lifted up with an acceleration of `4.9ms^(-2)` (ii) lowered with an acceleration of `4.9ms^(-2)`. The ratio of tension in the first and second case is

Options

A. `3:1`
B. `1:2`
C. `1:3`
D. `2:1`

Explanation

To solve the problem, we need to find the tension in the thread when a mass of 1 kg is lifted and lowered with an acceleration of 4.9 m/s². We will denote the tension when lifting as \( T_1 \) and the tension when lowering as \( T_2 \).

### Step 1: Calculate Tension When Lifting Up (T1)

When the mass is lifted with an acceleration \( a = 4.9 \, \text{m/s}^2 \), the forces acting on the mass are:
- The gravitational force acting downwards: \( F_g = mg = 1 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 9.8 \, \text{N} \)
- The tension \( T_1 \) acting upwards.

Using Newton's second law, we can write the equation for the mass when it is lifted:
\[ T_1 - F_g = ma \]
Substituting the values:
\[ T_1 - 9.8 = 1 \times 4.9 \]
\[ T_1 - 9.8 = 4.9 \]
\[ T_1 = 4.9 + 9.8 = 14.7 \, \text{N} \]

### Step 2: Calculate Tension When Lowering (T2)

When the mass is lowered with the same acceleration \( a = 4.9 \, \text{m/s}^2 \), the forces acting on the mass are:
- The gravitational force acting downwards: \( F_g = mg = 9.8 \, \text{N} \)
- The tension \( T_2 \) acting upwards.

Using Newton's second law for the case when it is lowered:
\[ T_2 - F_g = -ma \]
Substituting the values:
\[ T_2 - 9.8 = -1 \times 4.9 \]
\[ T_2 - 9.8 = -4.9 \]
\[ T_2 = 9.8 - 4.9 = 4.9 \, \text{N} \]

### Step 3: Calculate the Ratio of Tensions

Now we can find the ratio of the tensions \( T_1 \) and \( T_2 \):
\[ \text{Ratio} = \frac{T_1}{T_2} = \frac{14.7}{4.9} \]
Calculating the ratio:
\[ \text{Ratio} = 3 \]

### Final Answer
The ratio of tension in the first case (lifting) to the second case (lowering) is \( 3:1 \).

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