A 22.0 kg box is sliding down on an incline set at (44) degrees up from horizontal. The coefficient of kinetic friction between the box and the incline is (.27). Find the acceleration of the box. Give your answer in m/s2 and with 3 significant figures.
friction up plain: mu*mg*cosTheta
gravity down plane: mg*SinTheta
netforcedown=Mass* acceleration
gravity down-frictionup=m*a
solve for a.
To find the acceleration of the box, we need to analyze the forces acting on it.
1. Draw a diagram: Start by drawing a diagram of the situation. Label the incline, the box, and indicate the angle of the incline.
2. Identify the forces: There are several forces acting on the box:
- The force of gravity pulling the box downwards (mg), where g is the acceleration due to gravity (9.8 m/s²).
- The normal force (N) exerted by the incline on the box, perpendicular to the incline surface.
- The frictional force (f) opposing the motion of the box, parallel to the incline and directed up the slope.
3. Resolve the forces: Decompose the force of gravity into two components: one parallel to the incline (mg*sinθ) and one perpendicular to the incline (mg*cosθ), where θ is the angle of the incline (44 degrees).
- The perpendicular component (mg*cosθ) is balanced by the normal force (N) acting in the opposite direction.
- The parallel component (mg*sinθ) creates a force component that contributes to acceleration down the slope.
4. Calculate the force of friction: The frictional force (f) can be calculated using the formula f = μ*N, where μ is the coefficient of kinetic friction (0.27) and N is the normal force.
N = mg*cosθ
f = μ*N = μ*mg*cosθ
5. Determine the acceleration: The net force acting on the box is the component of gravity down the incline (mg*sinθ) minus the frictional force (f).
- Net force = mg*sinθ - f
Using Newton's second law (F = ma), we can set the net force equal to the product of mass (m) and acceleration (a), and solve for the acceleration:
mg*sinθ - f = ma
Substituting the value of f:
mg*sinθ - μ*mg*cosθ = ma
Now we can plug in the known values:
m = 22.0 kg
g = 9.8 m/s²
θ = 44 degrees
μ = 0.27
a = (mg*sinθ - μ*mg*cosθ) / m
Calculate the acceleration using these values, and round the answer to three significant figures.
Note: The acceleration will be negative because it opposes the downward motion of the box.