Someone with a mass of 86kg swings from a rope that is 14m long at angle of 45 degrees. The coefficient of friction is 0.5. How far did they slide?

To determine how far the person slid, we first need to analyze the energy involved in the swing and friction.

### Key points:
- **Mass (m):** 86 kg
- **Length of rope (L):** 14 m
- **Angle of swing (θ):** 45°
- **Coefficient of friction (μ):** 0.5
- We assume the sliding happens when the person lands after swinging.

### Step 1: Determine the potential energy at the highest point of the swing
At the start of the swing, the person is at a height \( h \) above the lowest point of the swing. We calculate the vertical height using trigonometry:
\[
h = L - L \cos(θ)
\]
\[
h = 14 - 14 \cos(45^\circ) = 14 - 14 \times 0.7071 = 14 - 9.8994 \approx 4.1 \, \text{m}
\]

### Step 2: Calculate the gravitational potential energy
The gravitational potential energy (PE) at the highest point is given by:
\[
PE = mgh
\]
Where:
- \( m = 86 \, \text{kg} \) (mass),
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity),
- \( h = 4.1 \, \text{m} \).

\[
PE = 86 \times 9.8 \times 4.1 \approx 3454.28 \, \text{J}
\]

### Step 3: Energy converted to kinetic energy
As the person swings down, all this potential energy is converted into kinetic energy (ignoring air resistance) at the lowest point of the swing:
\[
KE = \frac{1}{2} mv^2
\]
So, \( KE = 3454.28 \, \text{J} \).

### Step 4: Work done by friction (to stop the person)
When the person lands, they slide along the ground. The frictional force does work to stop the motion. The work done by friction \( W_f \) is given by:
\[
W_f = \mu \cdot N \cdot d
\]
Where:
- \( \mu = 0.5 \) (coefficient of friction),
- \( N = mg \) is the normal force, and
- \( d \) is the distance slid.

The normal force \( N \) is simply the person’s weight:
\[
N = mg = 86 \times 9.8 = 842.8 \, \text{N}
\]

Thus, the work done by friction becomes:
\[
W_f = 0.5 \times 842.8 \times d = 421.4 \times d
\]

### Step 5: Set the work done by friction equal to the kinetic energy
The work done by friction will equal the kinetic energy (since friction brings the person to rest):
\[
421.4 \times d = 3454.28
\]

Solving for \( d \):
\[
d = \frac{3454.28}{421.4} \approx 8.2 \, \text{m}
\]

### Final Answer:
The person slid approximately **8.2 meters**.

---

Would you like further details or have any other questions?

Here are 5 related questions to explore further:
1. What happens if the coefficient of friction were increased to 0.6?
2. How would a change in the swing angle affect the distance slid?
3. What if the mass of the person was different, say 75 kg, how would that impact the result?
4. How does air resistance affect the energy conversion during the swing?
5. How can we calculate the velocity at the lowest point of the swing?

**Tip:** When solving problems involving friction, always remember to account for the normal force, which is the weight of the object if it's sliding horizontally.

Someone with a mass of 86kg swings from a rope that is 14m long at an angle of 45 degrees. The coefficient of friction is 0.5. How far did they slide?

Learn how to calculate the distance slid after swinging from a 14m rope at a 45-degree angle, considering a mass of 86kg and a friction coefficient of 0.5. This problem covers energy conservation, potential and kinetic energy calculations, and frictional work. Suitable for students in Grades 10-12.

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