A block of mass 4kg with an initial kinetic energy of 600J ascends an incline tilted at 15 degrees to the horizontal. If the work done by friction between the block and the incline is -200J, how far does the block travel before coming to a rest?
To find the distance traveled by the block before it comes to a rest, we can use the work-energy principle. The principle states that the work done on an object is equal to the change in its kinetic energy.
Let's break down the problem step by step:
1. Given data:
- Mass of the block (m) = 4 kg
- Initial kinetic energy (KEi) = 600 J
- Angle of incline (θ) = 15 degrees
- Work done by friction (Wfriction) = -200 J
2. We know that the work done by friction is given by the equation:
Wfriction = -μ * m * g * d
where μ is the coefficient of friction, m is the mass of the block, g is the acceleration due to gravity, and d is the distance traveled by the block.
3. The gravitational force acting on the block can be broken down into its components:
- Weight of the block (mg) = m * g
- Component of weight parallel to the incline (mg * sin(θ)) = m * g * sin(θ)
4. The gravitational force parallel to the incline provides the force necessary to overcome friction. Therefore, the work done by the gravitational force is:
Wgravity = mg * sin(θ) * d
5. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy:
Wfriction + Wgravity = ΔKE
ΔKE = KEf - KEi
Since the block comes to a rest, its final kinetic energy (KEf) is zero.
6. Substituting the values and rearranging the equation, we get:
-200 J + (m * g * sin(θ) * d) = 0
7. Solving for d, we have:
d = 200 J / (m * g * sin(θ))
Now, let's plug in the values and calculate the distance traveled:
d = 200 J / (4 kg * 9.8 m/s^2 * sin(15 degrees))
d ≈ 3.09 meters
Therefore, the block travels approximately 3.09 meters before coming to a rest.