[9.] A 6.0 kg package in an Amazon sorting room slides 3.0 m down a plane inclined 60 degrees below the horizontal. The coefficient of kinetic friction between the package and the plane is k = 0.35. Calculate the work done on the package by (a) friction; (b) gravity; (c) the normal force. If the package starts with velocity v = 2.0 m/s, what is its final velocity?
[9.] A 6.0 kg package in an Amazon sorting room slides 3.0 m down a plane inclined 60 degrees below the horizontal. The coefficient of kinetic friction between the package and the plane is k = 0.35. Calculate the work done on the package by (a) friction; (b) gravity; (c) the normal force. If the package starts with velocity v = 2.0 m/s, what is its final velocity?
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Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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![**Physics Problem on Inclined Planes**
**Problem Statement:**
A 6.0 kg package in an Amazon sorting room slides 3.0 m down a plane inclined 60 degrees below the horizontal. The coefficient of kinetic friction between the package and the plane is \( \mu_k = 0.35 \). Calculate the work done on the package by (a) friction; (b) gravity; (c) the normal force. If the package starts with velocity \( v = 2.0 \, \text{m/s} \), what is its final velocity?
**Solution Overview:**
To solve this problem, we'll break it down into several steps:
1. **Determine the Forces:**
- Gravitational Force (Component along the incline).
- Frictional Force.
- Normal Force.
2. **Calculate the Work Done:**
- By friction.
- By gravity.
- By the normal force.
3. **Apply Energy Principles:**
- Use work-energy theorem to find the final velocity of the package.
### Detailed Steps:
**1. Gravitational Force (Component along the incline):**
- The gravitational force acting down the incline is given by:
\[ F_{\text{gravity, parallel}} = mg \sin \theta \]
- Where \( m = 6.0 \, \text{kg} \), \( g = 9.8 \, \text{m/s}^2 \), and \( \theta = 60^\circ \).
**2. Frictional Force:**
- The frictional force acting up the incline is calculated as:
\[ F_{\text{friction}} = \mu_k F_{\text{normal}} \]
- Where \( F_{\text{normal}} = mg \cos \theta \).
**3. Work Calculation:**
- **Work done by friction** \( W_{\text{friction}} \):
\[ W_{\text{friction}} = F_{\text{friction}} \times d \times \cos 180^\circ \]
- **Work done by gravity** \( W_{\text{gravity}} \):
\[ W_{\text{gravity}} = F_{\text{gravity, parallel}} \times d \times \cos 0^\circ \]
- **Work done by normal force** \( W_{\text{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba4cba5c-1d49-41b8-aac4-ce3ae0c647d3%2Fceb92204-eb53-4f6f-9ff4-fb47e6290309%2Fdsc9l8i_processed.png&w=3840&q=75)
Transcribed Image Text:**Physics Problem on Inclined Planes**
**Problem Statement:**
A 6.0 kg package in an Amazon sorting room slides 3.0 m down a plane inclined 60 degrees below the horizontal. The coefficient of kinetic friction between the package and the plane is \( \mu_k = 0.35 \). Calculate the work done on the package by (a) friction; (b) gravity; (c) the normal force. If the package starts with velocity \( v = 2.0 \, \text{m/s} \), what is its final velocity?
**Solution Overview:**
To solve this problem, we'll break it down into several steps:
1. **Determine the Forces:**
- Gravitational Force (Component along the incline).
- Frictional Force.
- Normal Force.
2. **Calculate the Work Done:**
- By friction.
- By gravity.
- By the normal force.
3. **Apply Energy Principles:**
- Use work-energy theorem to find the final velocity of the package.
### Detailed Steps:
**1. Gravitational Force (Component along the incline):**
- The gravitational force acting down the incline is given by:
\[ F_{\text{gravity, parallel}} = mg \sin \theta \]
- Where \( m = 6.0 \, \text{kg} \), \( g = 9.8 \, \text{m/s}^2 \), and \( \theta = 60^\circ \).
**2. Frictional Force:**
- The frictional force acting up the incline is calculated as:
\[ F_{\text{friction}} = \mu_k F_{\text{normal}} \]
- Where \( F_{\text{normal}} = mg \cos \theta \).
**3. Work Calculation:**
- **Work done by friction** \( W_{\text{friction}} \):
\[ W_{\text{friction}} = F_{\text{friction}} \times d \times \cos 180^\circ \]
- **Work done by gravity** \( W_{\text{gravity}} \):
\[ W_{\text{gravity}} = F_{\text{gravity, parallel}} \times d \times \cos 0^\circ \]
- **Work done by normal force** \( W_{\text{
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