On an essentially frictionless and inclined slope covered with ice sheet, a skier moving at 3.0m/s encounters a rough patch that lessens her acceleration due to friction force that is 25% of her weight.
To find the acceleration of the skier on the rough patch, we first need to calculate the friction force acting on her.
The friction force (F_friction) experienced by the skier can be calculated using the formula:
F_friction = μ * N
where μ is the coefficient of friction and N is the normal force.
In this case, the problem states that the friction force is 25% of the skier's weight. So we have:
F_friction = 0.25 * weight
Now let's calculate the normal force (N) acting on the skier.
On an inclined slope, the weight of an object can be resolved into two components: the component parallel to the slope direction (mg * sinθ) and the component perpendicular to the slope direction (mg * cosθ).
Since the slope is frictionless and essentially inclined, the only force acting perpendicular to the slope is the normal force (N), which is equal in magnitude and opposite in direction to the perpendicular component of the weight. So we have:
N = mg * cosθ
Now we can substitute this value into the equation for the friction force:
F_friction = 0.25 * weight
F_friction = 0.25 * (mg * cosθ)
Now we have the equation for the friction force in terms of the skier's weight and the angle of the slope. However, to further solve for the acceleration, we need to take into account the net force acting on the skier.
The net force (F_net) acting on the skier can be calculated using Newton's second law:
F_net = m * a
where m is the mass of the skier and a is the acceleration.
On an inclined plane, the net force can be resolved into two components: the component parallel to the slope (mg * sinθ) acting downward and the component perpendicular to the slope (F_friction) acting upward. So we have:
F_net = mg * sinθ - F_friction
Now we can substitute the value of F_friction obtained earlier:
F_net = mg * sinθ - 0.25 * (mg * cosθ)
Finally, we can solve for the acceleration by rearranging Newton's second law equation:
F_net = m * a
a = (F_net) / m
Substituting the value of F_net:
a = (mg * sinθ - 0.25 * (mg * cosθ)) / m
Note: The given initial velocity of the skier (3.0 m/s) is not needed to find the acceleration on the rough patch, as it only affects the skier's velocity and not the forces acting on her.