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To determine the **force of friction** (\( F_f \)) on the sled, we can use the following steps:

### Given Data:
- Mass of sled, \( m = 17.6 \, \text{kg} \)
- Applied force, \( F = 25.7 \, \text{N} \)
- Angle of applied force, \( \theta = 22.4^\circ \)
- Coefficient of kinetic friction, \( \mu_k = 0.135 \)

### Solution

1. **Calculate the Normal Force (\( F_N \)):**
   Since the force is applied at an angle, the vertical component of the applied force will reduce the normal force. The normal force is given by:
   \[
   F_N = mg - F \sin(\theta)
   \]
   where \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity).

2. **Calculate the Force of Friction (\( F_f \)):**
   The force of friction is given by:
   \[
   F_f = \mu_k \cdot F_N
   \]

### Step-by-Step Calculation:

#### Step 1: Calculate \( F_N \)
\[
F_N = (17.6 \, \text{kg})(9.81 \, \text{m/s}^2) - (25.7 \, \text{N}) \sin(22.4^\circ)
\]
1. Calculate \( mg \):
   \[
   mg = 17.6 \times 9.81 = 172.656 \, \text{N}
   \]
2. Calculate \( F \sin(\theta) \):
   \[
   F \sin(22.4^\circ) = 25.7 \times \sin(22.4^\circ) \approx 9.78 \, \text{N}
   \]
3. Substitute back to find \( F_N \):
   \[
   F_N = 172.656 - 9.78 = 162.876 \, \text{N}
   \]

#### Step 2: Calculate \( F_f \)
\[
F_f = \mu_k \cdot F_N = 0.135 \times 162.876 \approx 21.99 \, \text{N}
\]

### Answer:
The force of friction on the sled is approximately:
\[
F_f = -21.99 \, \text{N}
\]
(negative because friction opposes the direction of motion).

Would you like more detailed steps or have any questions?

### Follow-up Questions:
1. How would the friction force change if the angle of applied force increased?
2. What would happen if the coefficient of friction was zero?
3. How does the mass of the sled affect the normal force?
4. What is the role of the vertical component of the applied force in this scenario?
5. How would the friction force change if the sled was on an incline?

### Tip:
Always break forces into components when they’re applied at an angle, as it helps in calculating the normal force and friction correctly.

A 17.6 kg sled is pulled with a 25.7 N force at a 22.4° angle across the ground where the coefficient of friction (μk) is equal to 0.135. What is the force of friction on the sled?

Learn how to calculate the force of friction on a 17.6 kg sled pulled with a 25.7 N force at a 22.4° angle, given a coefficient of friction of 0.135. This problem involves trigonometry, friction, and force components, suitable for Grades 10-12.

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