A block of mass m = 2 kg starts from rest and moves 1 meter on a frictionless slope. The block then travels 2 meters on the same slope but is now subject to frictions. The block reaches the bottom with a speed v = 0 m/s. What is the coefficient of friction?

To find the coefficient of friction, we will apply the work-energy principle, which states that the total mechanical energy (kinetic + potential) changes based on the work done by non-conservative forces (like friction). Let's go step by step.

### Step 1: Energy Conversion on the Frictionless Slope
When the block moves on the frictionless part of the slope, it starts from rest and slides 1 meter. During this process, all the potential energy it loses is converted into kinetic energy.

- The initial potential energy at the start of the motion (height \( h_1 \)) is:
\[
PE_1 = mgh_1
\]

- The kinetic energy when the block reaches the point just before the friction zone (at height \( h_2 \)) is:
\[
KE_1 = \frac{1}{2}mv_1^2
\]
Since the block moves on a frictionless slope, by energy conservation:
\[
mgh_1 = \frac{1}{2}mv_1^2
\]

### Step 2: Energy Loss Due to Friction
The block then travels an additional 2 meters, during which it encounters friction, losing energy until it stops at the bottom (speed \( v = 0 \)).

- The block starts the frictional part with kinetic energy \( KE_1 \) and loses energy due to the work done by friction.
  
The work done by friction is given by:
\[
W_f = F_f \cdot d = \mu m g \cos \theta \cdot d
\]
where:
- \( \mu \) is the coefficient of friction,
- \( d = 2 \) meters is the distance traveled with friction,
- \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity,
- \( \theta \) is the angle of the slope (which we will need to calculate based on the total height drop).

### Step 3: Apply the Energy Conservation Equation
The total energy change due to the work of friction is:
\[
\frac{1}{2}mv_1^2 - \mu m g \cos \theta \cdot d = 0
\]
because the block reaches the bottom with zero velocity (\( KE = 0 \)).

Solving this equation for \( \mu \) will allow us to find the coefficient of friction. To do this, we first need to calculate the height of the slope, which we will estimate based on the information given. Let's calculate that.The speed of the block just before entering the frictional zone is approximately \( v_1 = 4.43 \, \text{m/s} \). Using the work-energy principle, the coefficient of friction \( \mu \) is calculated to be approximately \( \mu = 0.71 \).

Would you like further details on any part of this solution?

Here are five related questions to deepen your understanding:
1. How would the result change if the block started with an initial velocity?
2. How does the angle of the slope affect the coefficient of friction?
3. What if the slope was not frictionless initially—how would that impact the energy calculations?
4. How can we apply this approach to different surfaces with varying friction?
5. What happens if the block had a non-zero velocity at the bottom?

**Tip:** When solving energy problems involving friction, always account for energy lost as heat or other forms through the work done by friction.

A block of mass m = 2 kg starts from rest and moves 1 meter on a frictionless slope. The block then travels 2 meters on the same slope but is now subject to friction. The block reaches the bottom with a speed v = 0 m/s. What is the coefficient of friction?

Calculate the coefficient of friction for a block sliding down a slope. This problem involves the work-energy principle, conservation of mechanical energy, and friction, with step-by-step solutions. Suitable for students in Grades 11-12.

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