Sure, let's break down the problem step-by-step to find the maximum value of the external force [tex]\( e \)[/tex] for which the object remains at rest on the plane.
### Step 1: Mass and Gravity
Firstly, we are given the mass ([tex]\( m \)[/tex]) of the object:
[tex]\[ m = 5 \, \text{kg} \][/tex]
The acceleration due to gravity ([tex]\( g \)[/tex]) is a standard value:
[tex]\[ g = 9.8 \, \text{m/s}^2 \][/tex]
### Step 2: Calculate the Normal Force.
The normal force ([tex]\( F_{\text{normal}} \)[/tex]) is the force exerted by a surface to support the weight of an object resting on it. It is calculated as:
[tex]\[ F_{\text{normal}} = m \cdot g \][/tex]
Using the given values:
[tex]\[ F_{\text{normal}} = 5 \, \text{kg} \times 9.8 \, \text{m/s}^2 \][/tex]
[tex]\[ F_{\text{normal}} = 49.0 \, \text{N} \][/tex]
### Step 3: Coefficient of Static Friction
The coefficient of static friction ([tex]\( \mu_s \)[/tex]) is given as:
[tex]\[ \mu_s = 0.2 \][/tex]
### Step 4: Calculate the Maximum Force of Static Friction
The maximum force of static friction ([tex]\( F_{\text{static, max}} \)[/tex]) is the product of the coefficient of static friction and the normal force:
[tex]\[ F_{\text{static, max}} = \mu_s \cdot F_{\text{normal}} \][/tex]
Substitute the values into the equation:
[tex]\[ F_{\text{static, max}} = 0.2 \times 49.0 \, \text{N} \][/tex]
[tex]\[ F_{\text{static, max}} = 9.8 \, \text{N} \][/tex]
### Step 5: Maximum External Force
The maximum value of the external force ([tex]\( e \)[/tex]) for which the object remains at rest is equal to the maximum force of static friction:
[tex]\[ e = F_{\text{static, max}} \][/tex]
Hence:
[tex]\[ e = 9.8 \, \text{N} \][/tex]
### Summary
- The normal force ([tex]\( F_{\text{normal}} \)[/tex]) is [tex]\( 49.0 \, \text{N} \)[/tex].
- The maximum force of static friction ([tex]\( F_{\text{static, max}} \)[/tex]) is [tex]\( 9.8 \, \text{N} \)[/tex].
- The maximum value of the external force ([tex]\( e \)[/tex]) for which the object remains at rest is [tex]\( 9.8 \, \text{N} \)[/tex].
Thus, the maximum value of [tex]\( e \)[/tex] for which the object remains at rest on the plane is [tex]\( 9.8 \, \text{N} \)[/tex].
