https://file.aiquickdraw.com/mathbot/file/1726930347652hyte5wkc.jpg

We are given a mass of \(3 \, \text{kg}\) being pulled up a plane inclined at \(20^\circ\) to the horizontal by a rope. The mass is accelerating at \(0.6 \, \text{m/s}^2\) up the incline, and the coefficient of friction between the mass and the plane is \(0.4\). We are tasked with finding the tension in the rope \(T\).

### Step-by-Step Solution

#### 1. Identify Forces Acting on the Block:
- Gravitational force \( F_g = mg \), where \( m = 3 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \).
- Normal force \( R \) perpendicular to the plane.
- Frictional force \( F_f = \mu R \), where \( \mu = 0.4 \).
- Tension \( T \) in the rope, pulling the mass up the plane.
- Component of gravity parallel to the plane \( F_{\parallel} = mg \sin \theta \).
- Component of gravity perpendicular to the plane \( F_{\perp} = mg \cos \theta \).

#### 2. Calculate the Gravitational Components:
- \( F_{\parallel} = 3 \times 9.8 \times \sin 20^\circ \)
- \( F_{\perp} = 3 \times 9.8 \times \cos 20^\circ \)

#### 3. Normal Force \(R\):
The normal force is equal to the perpendicular component of gravity:
\[
R = F_{\perp} = 3 \times 9.8 \times \cos 20^\circ
\]

#### 4. Frictional Force:
The frictional force is given by:
\[
F_f = \mu R = 0.4 \times R
\]

#### 5. Net Force:
The net force acting along the plane is the result of tension, gravitational force, and friction:
\[
F_{\text{net}} = T - F_f - F_{\parallel}
\]
We know the block is accelerating at \( a = 0.6 \, \text{m/s}^2 \), so:
\[
F_{\text{net}} = ma
\]
\[
T - F_f - F_{\parallel} = 3 \times 0.6
\]
Solve this for \( T \).

Let me calculate the values.Here are the calculated forces:

- The component of gravity parallel to the incline: \( F_{\parallel} = 10.06 \, \text{N} \).
- The component of gravity perpendicular to the incline: \( F_{\perp} = 27.63 \, \text{N} \).
- The frictional force: \( F_f = 11.05 \, \text{N} \).
- The tension in the rope: \( T = 22.91 \, \text{N} \).

Thus, the tension in the rope is approximately **22.91 N**.

Would you like further details or have any questions?

### Related Questions:
1. What would the tension be if the angle of inclination increased?
2. How would the tension change if the coefficient of friction was reduced to zero?
3. How does the normal force change with the angle of inclination?
4. What effect does increasing the mass of the block have on the tension?
5. How is the tension affected if the acceleration is doubled?

### Tip:
In incline problems, always decompose forces into components parallel and perpendicular to the plane for easier analysis.

A mass of 3 kg is being pulled up a plane inclined at 20 degrees to the horizontal by a rope parallel to the surface. The mass is accelerating at 0.6 m/s^2, and the coefficient of friction between the mass and the plane is 0.4. Find the tension in the rope.

Solve a problem involving a 3 kg mass being pulled up an inclined plane at 20 degrees. The mass is accelerating at 0.6 m/s^2 with friction (μ = 0.4). Learn how to calculate the tension in the rope using Newton's laws and force decomposition.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation