The block shown in Figure 6 has mass 3500 g and lies on a plane which tilted at an angle θ = 22° to the horizontal. The effective coefficient of kinetic friction is 0.20. Determine the acceleration of the block as it slides down the plane.
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To determine the acceleration of the block as it slides down the plane, we need to consider the forces acting on the block.
First, let's find the components of the weight force of the block. The weight force can be split into two components: the component parallel to the plane (mg*sinθ) and the component perpendicular to the plane (mg*cosθ), where m is the mass of the block and g is the acceleration due to gravity.
The parallel component of the weight force, mg*sinθ, will act down the plane. The perpendicular component of the weight force, mg*cosθ, will be balanced by the normal force from the plane.
Now, let's consider the friction force. The friction force can be calculated by multiplying the coefficient of kinetic friction (μk) by the normal force (mg*cosθ).
The net force (Fnet) acting on the block in the direction of motion down the plane can be calculated as follows:
Fnet = mg*sinθ - μk*(mg*cosθ)
Finally, we'll use Newton's second law, which states that the acceleration (a) is equal to the net force divided by the mass of the block (m):
a = Fnet / m
Plugging in the given values:
- Mass (m) = 3500 g = 3.5 kg
- Angle (θ) = 22°
- Coefficient of kinetic friction (μk) = 0.20
- Acceleration due to gravity (g) = 9.8 m/s²
We can now calculate the acceleration (a) using the above equations. Let me do the math.
Calculating the parallel component of the weight force:
mg*sinθ = (3.5 kg)(9.8 m/s²)(sin 22°) = 12.62 N
Calculating the perpendicular component of the weight force:
mg*cosθ = (3.5 kg)(9.8 m/s²)(cos 22°) = 32.14 N
Calculating the friction force:
friction force = μk*(mg*cosθ) = 0.20(32.14 N) = 6.43 N
Calculating the net force:
Fnet = mg*sinθ - friction force = 12.62 N - 6.43 N = 6.19 N
Calculating the acceleration:
a = Fnet / m = 6.19 N / 3.5 kg = 1.77 m/s²
Therefore, the acceleration of the block as it slides down the plane is 1.77 m/s².