A block has a mass m=6.9-kg and lies on a fixed smooth frictionless plane tilted at an angle of 24.4 to the horizontal. Determine the acceleration (in m/s2)of the block as it slides down the plane.
g sin(24.4)
To find the acceleration of the block as it slides down the plane, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.
Since the plane is smooth and frictionless, the only force acting on the block is its weight, which can be calculated as the product of its mass (m) and the acceleration due to gravity (g).
Finding the component of the weight parallel to the plane will give us the net force acting on the block.
1. Find the component of the weight parallel to the plane:
- The weight (W) of the block is given by:
W = m * g
where g is the acceleration due to gravity, approximately 9.8 m/s^2.
- The component of the weight parallel to the plane (W_parallel) can be found using the formula:
W_parallel = W * sin(θ)
where θ is the angle of the plane with respect to the horizontal.
2. Calculate the net force acting on the block:
- The net force (F_net) acting on the block is equal to the component of the weight parallel to the plane, W_parallel.
F_net = W_parallel
3. Use Newton's second law of motion to find the acceleration:
- Newton's second law states that the net force (F_net) acting on an object is equal to the product of its mass (m) and acceleration (a):
F_net = m * a
- Rearrange the equation to solve for the acceleration (a):
a = F_net / m
4. Substitute the values:
- Substitute the known values into the equations:
W_parallel = m * g * sin(θ)
a = W_parallel / m
- The given values are:
m = 6.9 kg (mass of the block)
θ = 24.4° (angle of the plane)
- Substitute the values to calculate the acceleration:
W_parallel = 6.9 kg * 9.8 m/s^2 * sin(24.4°)
a = (6.9 kg * 9.8 m/s^2 * sin(24.4°)) / 6.9 kg
5. Calculate the acceleration:
- Evaluate the expression:
a ≈ 3.42 m/s^2
Therefore, the acceleration of the block as it slides down the plane is approximately 3.42 m/s^2.