The block shown in figure below lies on a smooth plane tilted at an angle è = 21.5° to the horizontal. Determine the acceleration of the block as it slides down the plane. Ignore friction.
force down slope = m g sin 21 .5 = m a
To determine the acceleration of the block as it slides down the plane, you can use the following steps:
Step 1: Analyze the forces acting on the block.
There are two forces acting on the block:
- The gravitational force (mg), which can be split into two components: the force acting parallel to the inclined plane (mg sinθ) and the force acting perpendicular to the inclined plane (mg cosθ).
- The normal force (N) acting perpendicular to the inclined plane.
Step 2: Resolve the gravitational force.
The force acting parallel to the inclined plane (mg sinθ) is the force responsible for the block's acceleration. The force acting perpendicular to the inclined plane (mg cosθ) is balanced by the normal force (N).
Step 3: Apply Newton's second law.
Since there is no friction, the net force acting on the block is equal to the force parallel to the inclined plane (mg sinθ).
Step 4: Determine the acceleration.
Using Newton's second law (F = ma), we have:
mg sinθ = ma
Step 5: Solve for acceleration.
Rearrange the equation to solve for acceleration:
a = (mg sinθ) / m
a = g sinθ
Now we can calculate the acceleration using the given angle è = 21.5° and the acceleration due to gravity g = 9.8 m/s²:
a = 9.8 m/s² * sin(21.5°)
a ≈ 3.34 m/s²
Therefore, the acceleration of the block as it slides down the plane, ignoring friction, is approximately 3.34 m/s².
To determine the acceleration of the block as it slides down the plane, we need to analyze the forces acting on the block.
1. Gravitational Force: The weight of the block acts vertically downward. We can resolve this force into two components: one parallel to the incline and one perpendicular to the incline. The component parallel to the incline is responsible for the acceleration along the plane.
2. Normal Force: The surface of the inclined plane exerts a normal force perpendicular to the plane. Since there is no friction (as stated in the question), the normal force will be equal in magnitude to the perpendicular component of the weight.
3. Frictional Force (ignored): As mentioned in the question, friction is ignored, so we do not need to consider it here.
Now, let's break down the forces and determine the acceleration:
Step 1: Resolve the weight into components:
- Weight (W): W = m * g, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s²).
- Perpendicular Component (W⊥): W⊥ = W * cos(θ), where θ is the angle of inclination (21.5°).
- Parallel Component (W||): W|| = W * sin(θ).
Step 2: Determine the normal force (N):
- Since there is no vertical acceleration, the normal force (N) must balance out the perpendicular component of the weight: N = W⊥.
Step 3: Calculate the acceleration (a):
- The only force acting parallel to the incline is the parallel component of weight (W||).
- Since there is no friction, the net force (Fnet) parallel to the incline will be equal to W||.
- Using Newton's second law (Fnet = m * a), we can substitute W|| for Fnet: W|| = m * a.
- Finally, we can solve for the acceleration (a): a = W|| / m.
By following these steps and substituting the values of the given variables, you can calculate the acceleration of the block as it slides down the plane.