A skier of mass m is sliding down a hill that makes an angle 𝜃 with the horizontal. There is a frictional force of magnitude f resisting the motion of the skier. a) determine the magnitude of the force that the hill’s surface exerts on the skier.
b) Explain what happens to this force as 𝜃 increases.
c) Explain for what value of 𝜃 the force in question maximum.
a) To determine the magnitude of the force that the hill's surface exerts on the skier, we need to consider the forces acting on the skier along the direction of the slope.
The gravitational force, mg, can be decomposed into two components: mg sin 𝜃, acting downhill, and mg cos 𝜃, acting perpendicular to the slope.
The frictional force, f, is resisting the motion of the skier downhill. However, since the skier is already sliding down the hill, the frictional force is not enough to stop the skier's motion. Hence, we assume that the skier is sliding with a constant velocity.
The net force acting downhill is given by the difference between the downhill component of gravity and the frictional force:
Net Force = mg sin 𝜃 - f
Since the skier is sliding with a constant velocity, the net force must be zero. Therefore,
mg sin 𝜃 - f = 0
Solving for f, we have:
f = mg sin 𝜃
So, the magnitude of the force that the hill's surface exerts on the skier is mg sin 𝜃.
b) As 𝜃 increases, the component of the gravitational force acting downhill, mg sin 𝜃, also increases. This means that the force required to counteract this component, which is the force exerted by the hill's surface on the skier, also increases.
c) The force in question will be maximum when the angle 𝜃 is such that the downhill component of gravity, mg sin 𝜃, is equal to the maximum frictional force that can be exerted by the hill's surface. This maximum frictional force is given by the product of the coefficient of friction, μ, and the perpendicular component of gravity, mg cos 𝜃.
So, when mg sin 𝜃 = μmg cos 𝜃, the force is maximum. Simplifying, we have:
sin 𝜃 = μ cos 𝜃
Dividing both sides by cos 𝜃, we get:
tan 𝜃 = μ
Therefore, the value of 𝜃 for which the force in question is maximum is the angle at which the tangent of 𝜃 is equal to the coefficient of friction, μ.
a) To determine the magnitude of the force that the hill's surface exerts on the skier, we'll consider the forces acting on the skier. There are two main forces involved: the component of the gravitational force pulling the skier downhill, and the frictional force opposing the skier's motion.
Let's break down the forces acting on the skier parallel to the hill's surface. The downhill component of the gravitational force is given by mg*sin(𝜃), where m is the mass of the skier, g is the acceleration due to gravity, and 𝜃 is the angle of the hill with respect to the horizontal.
The frictional force opposing the skier's motion is given by f (as mentioned in the problem statement). For now, let's assume the skier is not accelerating. In this case, the downhill component of the gravitational force equals the frictional force:
mg*sin(𝜃) = f
Therefore, the magnitude of the force that the hill's surface exerts on the skier is equal to the frictional force, f.
b) As 𝜃 increases, the force that the hill's surface exerts on the skier decreases. This can be understood by looking at the downhill component of the gravitational force, mg*sin(𝜃). As 𝜃 increases, sin(𝜃) also increases. Since the downhill component of the gravitational force is proportional to sin(𝜃), it becomes larger as 𝜃 increases.
Meanwhile, the frictional force, f, remains constant. Thus, as the downhill component of the gravitational force becomes larger relative to the constant frictional force, the net force on the skier decreases. Consequently, the force that the hill's surface exerts on the skier decreases.
c) The force described in question b is maximum when 𝜃 = 90 degrees (or π/2 radians). At this point, the hill's surface is perpendicular to the horizontal, and the skier is essentially moving straight down the hill. In this case, the entire gravitational force mg acts parallel to the hill's surface, providing the maximum force that the hill's surface can exert on the skier.
Note that for 𝜃 > 90 degrees, the component of the gravitational force perpendicular to the hill's surface becomes negative, effectively causing the skier to push into the hill rather than being supported by it. Therefore, the force described in question b is only applicable for 𝜃 ≤ 90 degrees.