You are pushing a sled in which your little sister is seated up a 28° slope (one that makes an angle of 28° with the horizontal). If the mass of the sled and girl is 36 kg, and the coefficient of kinetic friction between the sled and the surface is 0.100, with what force (in N) do you need to push in order for the sled to move with constant velocity?

To determine the force required to push the sled up the slope with constant velocity, we need to consider the forces acting on the sled and use Newton's second law. Here are the steps to solve the problem:

1. **Identify the forces acting on the sled:**
   - Gravitational force (\( F_g \))
   - Normal force (\( F_N \))
   - Frictional force (\( F_f \))
   - Applied force (\( F_{\text{push}} \))

2. **Break down the gravitational force into components:**
   - Parallel to the slope (\( F_{g,\parallel} \))
   - Perpendicular to the slope (\( F_{g,\perp} \))

3. **Calculate the normal force:**
   - The normal force is the component of the gravitational force perpendicular to the slope.
  
4. **Calculate the frictional force:**
   - The frictional force is the product of the normal force and the coefficient of kinetic friction.

5. **Set up the equation for constant velocity:**
   - For constant velocity, the net force along the slope must be zero, meaning the applied force must counteract both the parallel component of the gravitational force and the frictional force.

Let's calculate each step:

### Gravitational Force Components
The gravitational force (\( F_g \)) is given by:
\[ F_g = mg \]
where \( m = 36 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \).

\[ F_g = 36 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 352.8 \, \text{N} \]

The component of the gravitational force parallel to the slope (\( F_{g,\parallel} \)) is:
\[ F_{g,\parallel} = F_g \sin(28^\circ) \]
\[ F_{g,\parallel} = 352.8 \, \text{N} \times \sin(28^\circ) \]
\[ F_{g,\parallel} \approx 352.8 \, \text{N} \times 0.4695 \]
\[ F_{g,\parallel} \approx 165.55 \, \text{N} \]

The component of the gravitational force perpendicular to the slope (\( F_{g,\perp} \)) is:
\[ F_{g,\perp} = F_g \cos(28^\circ) \]
\[ F_{g,\perp} = 352.8 \, \text{N} \times \cos(28^\circ) \]
\[ F_{g,\perp} \approx 352.8 \, \text{N} \times 0.8829 \]
\[ F_{g,\perp} \approx 311.34 \, \text{N} \]

### Normal Force
The normal force (\( F_N \)) is equal to the perpendicular component of the gravitational force:
\[ F_N = F_{g,\perp} \approx 311.34 \, \text{N} \]

### Frictional Force
The frictional force (\( F_f \)) is given by:
\[ F_f = \mu_k F_N \]
where \( \mu_k = 0.100 \).

\[ F_f = 0.100 \times 311.34 \, \text{N} \]
\[ F_f \approx 31.13 \, \text{N} \]

### Applied Force for Constant Velocity
To move the sled with constant velocity, the applied force must counteract both the parallel component of the gravitational force and the frictional force:

\[ F_{\text{push}} = F_{g,\parallel} + F_f \]
\[ F_{\text{push}} = 165.55 \, \text{N} + 31.13 \, \text{N} \]
\[ F_{\text{push}} \approx 196.68 \, \text{N} \]

So, the force required to push the sled up the slope with constant velocity is approximately **196.68 N**.

Do you want details or have any questions?

Here are some more questions you might find interesting:
1. How to calculate the work done by a force on an inclined plane?
2. What is the effect of increasing the angle of the slope on the required force?
3. How to determine the acceleration if the applied force is not balanced?
4. What are the differences between static and kinetic friction?
5. How does the normal force change with different angles of the slope?

**Tip:** When solving physics problems involving forces on slopes, always resolve the forces into components parallel and perpendicular to the slope for easier analysis.

Learn how to calculate the force needed to push a sled up a 28° slope with constant velocity. This physics problem involves Newton's Laws of Motion, forces on inclined planes, gravitational force, frictional force, and coefficient of kinetic friction. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation