A small body of mass m is located on a horizontal plane at the point O. The body acquires a horizontal velocity `v_0`. Find the mean power developed by the friction force during the whole time of motion, if the frictional coefficient `mu=0.27`,`m=1.0kg` and `v_0=1.5ms^-1`.

Options

A. `-mu mg v_(0)`
B. `(-mu mg v_(0))/(2)`
C. `(-mu mg v_(0))/(4)`
D. `(-mu mg v_(0))/(8)`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the forces acting on the body
The only horizontal force acting on the body is the frictional force, which opposes the motion. The frictional force \( F_f \) can be calculated using the formula:
\[ F_f = \mu \cdot m \cdot g \]
where:
- \( \mu = 0.27 \) (coefficient of friction)
- \( m = 1.0 \, \text{kg} \) (mass of the body)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)

### Step 2: Calculate the frictional force
Substituting the values into the equation:
\[ F_f = 0.27 \cdot 1.0 \cdot 10 = 2.7 \, \text{N} \]

### Step 3: Determine the acceleration due to friction
Using Newton's second law, the acceleration \( a \) caused by the frictional force can be calculated as:
\[ a = \frac{F_f}{m} = \frac{\mu \cdot m \cdot g}{m} = \mu \cdot g \]
Substituting the values:
\[ a = 0.27 \cdot 10 = 2.7 \, \text{m/s}^2 \]

### Step 4: Calculate the time taken to stop
Using the formula for time \( t \) when initial velocity \( v_0 \) and acceleration \( a \) are known:
\[ t = \frac{v_0}{a} \]
Substituting the values:
\[ t = \frac{1.5}{2.7} \approx 0.5556 \, \text{s} \]

### Step 5: Calculate the work done by the friction force
The work done \( W \) by the friction force can be calculated using the formula:
\[ W = F_f \cdot d \]
where \( d \) is the distance traveled before coming to rest. The distance can be calculated using:
\[ d = \frac{v_0^2}{2a} \]
Substituting the values:
\[ d = \frac{(1.5)^2}{2 \cdot 2.7} = \frac{2.25}{5.4} \approx 0.4167 \, \text{m} \]
Now, substituting \( d \) back into the work done equation:
\[ W = F_f \cdot d = 2.7 \cdot 0.4167 \approx 1.125 \, \text{J} \]

### Step 6: Calculate the mean power developed by the friction force
Mean power \( P \) can be calculated using the formula:
\[ P = \frac{W}{t} \]
Substituting the values:
\[ P = \frac{1.125}{0.5556} \approx 2.02 \, \text{W} \]

### Final Answer
Thus, the mean power developed by the friction force during the whole time of motion is approximately \( 2.02 \, \text{W} \).

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