A 69 kg skier speeds down a trail, as shown in the figure. The surface is smooth and inclined at an angle of θ = 17° with the horizontal.
Help me with an equation to solve. thanks!
(a) Find the direction and magnitude of the net force acting on the skier.
Magnitude
N Direction
perpendicular from the slope (into the ground) perpendicular from the slope (away from ground) downhill, parallel to slope away from the center of the earth towards the center of the earth uphill, parallel to slope
(b) Does the net force exerted on the skier increase, decrease, or stay the same as the slope becomes steeper?
stay the same decrease increase
To solve this problem, we can consider the forces acting on the skier. Let's break the force vector into components.
The force of gravity acting on the skier can be resolved into two components:
- The component parallel to the incline, which is mg*sin(θ), where m is the mass of the skier and g is the acceleration due to gravity.
- The component perpendicular to the incline, which is mg*cos(θ).
Since the surface is smooth, there is no friction acting on the skier, so we can ignore that force.
Now, let's consider the acceleration of the skier along the incline. We can use Newton's second law, which states that the net force acting on an object is equal to its mass (m) times its acceleration (a):
Net force = m * a
The net force acting on the skier is in the direction parallel to the incline and is given by:
Net force = m * g * sin(θ)
Since the net force is responsible for the acceleration of the skier, we can equate these two expressions:
m * g * sin(θ) = m * a
Now, we can solve this equation for the acceleration (a):
a = g * sin(θ)
So, the equation to solve for the acceleration of the skier is a = g * sin(θ), where g is the acceleration due to gravity (approximately 9.8 m/s^2) and θ is the angle of the incline (17°).
To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the net force acting on the skier can be broken down into two components: the force due to gravity pulling the skier downhill and the force due to the friction between the skis and the surface.
The force due to gravity can be calculated using the equation Fg = m * g, where m is the mass of the skier (69 kg) and g is the acceleration due to gravity (9.8 m/s^2).
The force due to friction can be calculated using the equation Ff = μ * Fn, where μ is the coefficient of friction and Fn is the normal force acting on the skier. The normal force is the force exerted by the surface perpendicular to the skier and can be calculated as Fn = m * g * cos(θ), where θ is the angle of the incline (17°).
The net force acting on the skier is the vector sum of the force due to gravity and the force due to friction. In this case, the net force is acting downhill, parallel to the incline.
Therefore, the equation to solve for the acceleration of the skier is:
m * a = m * g * sin(θ) - μ * m * g * cos(θ)
Once you have the acceleration, you can use it to find other quantities such as the skier's speed at any given point, using kinematic equations.