An 8.65-kg block slides with an initial speed of 1.52 m/s up a ramp inclined at an angle of 27.6° with the horizontal. The coefficient of kinetic friction between the block and the ramp is 0.56. Use energy conservation to find the distance the block slides before coming to rest.
To find the distance the block slides before coming to rest, we can use the principle of conservation of mechanical energy. Initially, the block has kinetic energy due to its motion, and as it slides up the ramp, this kinetic energy is gradually converted into other forms, such as potential energy and work done against friction.
1. Identify the initial and final points: The initial point is when the block starts sliding, and the final point is when the block comes to rest.
2. Determine the initial and final energies: The initial energy is the kinetic energy of the block, given by the formula: KE = 0.5 * m * v², where m is the mass of the block and v is its initial velocity. The final energy is zero since the block comes to rest.
3. Find the work done against friction: The work done against friction can be calculated using the formula: Work = force of friction * distance, where the force of friction is the product of the coefficient of kinetic friction (μk) and the normal force. The normal force can be calculated as: N = m * g * cos(θ), where m is the mass of the block, g is the acceleration due to gravity (approximately 9.8 m/s²), and θ is the angle of the ramp with the horizontal.
4. Apply the law of conservation of mechanical energy: The initial kinetic energy minus the work done against friction must be equal to the final energy. Mathematically:
0.5 * m * v² - Work = 0
5. Rearrange the equation and solve for distance: Rearrange the equation to solve for the distance: distance = Work / force of friction.
6. Calculate the final answer: Substitute the known values into the equation and solve for the distance.
Here are the detailed calculations:
Given data:
- Mass of the block (m) = 8.65 kg
- Initial velocity (v) = 1.52 m/s
- Angle of the ramp (θ) = 27.6°
- Coefficient of kinetic friction (μk) = 0.56
- Acceleration due to gravity (g) = 9.8 m/s²
Step 2: Initial kinetic energy
KE = 0.5 * m * v² = 0.5 * 8.65 kg * (1.52 m/s)² = 10.042 J
Step 3: Work done against friction
N (normal force) = m * g * cos(θ) = 8.65 kg * 9.8 m/s² * cos(27.6°) = 76.15 N
force of friction = μk * N = 0.56 * 76.15 N = 42.616 N
Step 5: Distance calculation
distance = Work / force of friction
Step 6: Final answer
Substituting the values into the equation, we get:
distance = Work / force of friction = 10.042 J / 42.616 N ≈ 0.236 m
Therefore, the block slides approximately 0.236 meters before coming to rest.