To solve the problem, we will analyze the two scenarios separately:
### Part (a): Lift moving upwards with a uniform retardation of \(2.8 \, \text{m/s}^2\)
1. **Identify the forces acting on the person**:
- The weight of the person acting downwards: \( W = mg \)
- The normal force (reading of the weighing machine) acting upwards: \( R \)
2. **Given data**:
- Mass of the person, \( m = 60 \, \text{kg} \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)
- Retardation (negative acceleration), \( a = -2.8 \, \text{m/s}^2 \)
3. **Calculate the weight of the person**:
\[
W = mg = 60 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 588 \, \text{N}
\]
4. **Set up the equation of motion**:
Since the lift is moving upwards with retardation, we take upward as positive. The equation of motion is:
\[
R - W = ma
\]
Substituting \( a = -2.8 \, \text{m/s}^2 \):
\[
R - mg = m(-2.8)
\]
Rearranging gives:
\[
R = mg + m(-2.8)
\]
\[
R = m(g - a)
\]
5. **Substitute the values**:
\[
R = 60 \, \text{kg} \times (9.8 \, \text{m/s}^2 - 2.8 \, \text{m/s}^2)
\]
\[
R = 60 \, \text{kg} \times 7.0 \, \text{m/s}^2 = 420 \, \text{N}
\]
### Part (b): Lift moving downwards with a uniform retardation of \(2.2 \, \text{m/s}^2\)
1. **Identify the forces acting on the person**:
- The weight of the person acting downwards: \( W = mg \)
- The normal force (reading of the weighing machine) acting upwards: \( R \)
2. **Given data**:
- Mass of the person, \( m = 60 \, \text{kg} \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)
- Retardation (negative acceleration), \( a = -2.2 \, \text{m/s}^2 \)
3. **Calculate the weight of the person**:
\[
W = mg = 60 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 588 \, \text{N}
\]
4. **Set up the equation of motion**:
Since the lift is moving downwards with retardation, we take downward as positive. The equation of motion is:
\[
mg - R = ma
\]
Rearranging gives:
\[
R = mg - ma
\]
\[
R = m(g + a)
\]
5. **Substitute the values**:
\[
R = 60 \, \text{kg} \times (9.8 \, \text{m/s}^2 + 2.2 \, \text{m/s}^2)
\]
\[
R = 60 \, \text{kg} \times 12.0 \, \text{m/s}^2 = 720 \, \text{N}
\]
### Final Answers:
- For part (a): The reading shown by the weighing machine is \( 420 \, \text{N} \).
- For part (b): The reading shown by the weighing machine is \( 720 \, \text{N} \).
