A block of mass "`5kg`" is placed on a rough horizontal surface . A veriable force of `5t` Newton acts on it horizontally . If block starts slipping at `t=4 sec` and its acceletation at `t=5 sec` is `2 m//s^(2)`.Find the value of `10(mu_(s)-mu_(k))`.

Options

A. `1`
B. `2`
C. `3`
D. `4`

Explanation

To solve the problem step by step, we need to analyze the forces acting on the block and use the given information to find the value of \(10(\mu_s - \mu_k)\).

### Step 1: Understand the Forces Acting on the Block
The block of mass \(m = 5 \, \text{kg}\) is placed on a rough horizontal surface. The forces acting on the block include:
- The applied force \(F = 5t \, \text{N}\)
- The weight of the block \(W = mg = 5 \times 9.8 = 49 \, \text{N}\)
- The normal force \(N\) which balances the weight, hence \(N = 49 \, \text{N}\)
- The frictional force which is dependent on the coefficients of static and kinetic friction.

### Step 2: Determine the Static Friction Coefficient
The block starts slipping at \(t = 4 \, \text{s}\). At this moment, the applied force equals the maximum static friction force:
\[ F = \mu_s N \]
Substituting the values:
\[ 5 \times 4 = \mu_s \times 50 \]
\[ 20 = \mu_s \times 50 \]
Solving for \(\mu_s\):
\[ \mu_s = \frac{20}{50} = \frac{2}{5} \]

### Step 3: Determine the Kinetic Friction Coefficient
At \(t = 5 \, \text{s}\), the block is in motion, and we can use the acceleration given to find the kinetic friction coefficient. The net force acting on the block is:
\[ F_{\text{net}} = F - F_k \]
Where \(F_k = \mu_k N\). The net force can also be expressed using Newton's second law:
\[ F_{\text{net}} = ma \]
Substituting the known values:
\[ 5t - \mu_k N = ma \]
At \(t = 5 \, \text{s}\) and \(a = 2 \, \text{m/s}^2\):
\[ 5 \times 5 - \mu_k \times 50 = 5 \times 2 \]
This simplifies to:
\[ 25 - 50\mu_k = 10 \]
Rearranging gives:
\[ 50\mu_k = 25 - 10 = 15 \]
Thus,
\[ \mu_k = \frac{15}{50} = \frac{3}{10} \]

### Step 4: Calculate \(10(\mu_s - \mu_k)\)
Now that we have both coefficients:
\[ \mu_s = \frac{2}{5} = \frac{4}{10}, \quad \mu_k = \frac{3}{10} \]
We can find \(10(\mu_s - \mu_k)\):
\[ 10(\mu_s - \mu_k) = 10\left(\frac{4}{10} - \frac{3}{10}\right) = 10 \times \frac{1}{10} = 1 \]

### Final Answer
The value of \(10(\mu_s - \mu_k)\) is \(1\).