A boy is sledding down a slope that is inclined at 30.0° with respect to the horizontal. A moderate wind is aiding the motion by providing a steady force of 108 N that is parallel to the motion of the sled. The combined mass of the boy and sled is 53.6 kg, and the coefficient of friction between the runners of the sled and the snow is 0.150.
What is the acceleration of the boy and his sled?
x: m•a= m•g•sinα +F - F(fr),
y: 0= - m•g•cosα +N.
F(fr)=μ•N= μ• m•g•cosα
m•a= m•g•sinα +F - μ• m•g•cosα
a= g(sinα - μ•cosα)+ F
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To find the acceleration of the boy and his sled, we need to consider the forces acting on them and apply Newton's second law of motion.
The forces acting on the boy and sled are:
1. The force of gravity acting vertically downwards, which can be split into two components: the force parallel to the incline (mg * sinθ) and the force perpendicular to the incline (mg * cosθ).
2. The applied force by the wind, which is parallel to the motion of the sled (108 N).
3. The force of friction opposing the motion, which can be calculated as the coefficient of friction (μ) multiplied by the normal force (mg * cosθ).
Since we are interested in the acceleration, we can consider the forces parallel to the incline only. Let's define the positive direction as down the slope.
The net force acting parallel to the incline (F_net) can be expressed as:
F_net = (mg * sinθ) + (108 N) - (μ * (mg * cosθ))
Now, we can apply Newton's second law, which states that the net force acting on an object is equal to its mass (m) multiplied by its acceleration (a):
F_net = m * a
By substituting the values and solving for acceleration, we can find the answer. Let's plug in the known values:
(mg * sinθ) + (108 N) - (μ * (mg * cosθ)) = (m * a)
Given values:
m = 53.6 kg
g = 9.8 m/s^2 (acceleration due to gravity)
θ = 30.0°
μ = 0.150
Calculating the acceleration:
(a * m) = (mg * sinθ) + (108 N) - (μ * (mg * cosθ))
a = [(mg * sinθ) + (108 N) - (μ * (mg * cosθ))] / m
Substituting the known values, we can find the final answer. Doing the calculation, the acceleration of the boy and his sled is approximately -0.0917 m/s^2. The negative sign indicates that the acceleration is directed up the slope, which means it opposes the motion down the slope.