A snowboarder of mass 60 kg is at rest on

the top of a sand dune. The coefficients of
static and kinetic friction are 0.55 and 0.4675,
respectively.

60 kg
μk = 0.4675
μs = 0.55

What is the angle � the incline must exceed
so that the snowboarder starts sliding? The
acceleration of gravity is 9.8 m/s2 .
Answer in units of ◦.

I worked out the variables but don't know where to go from there.

Well, it seems like you're stuck in a slippery situation! Don't worry, I'll lead you down the right slope.

To determine the angle of the incline that the snowboarder must exceed to start sliding, we need to consider the forces acting on the snowboarder.

First, let's calculate the force of gravity acting on the snowboarder using the mass and acceleration due to gravity:

Force of gravity = mass × acceleration due to gravity

Fg = (60 kg) × (9.8 m/s^2)

Now, we need to calculate the maximum frictional force that can be exerted by the static friction:

Maximum static frictional force = coefficient of static friction × normal force

Since the snowboarder is at rest, the static friction must balance the force of gravity. Therefore, the normal force is equal to the force of gravity:

Normal force = Force of gravity

Next, we can calculate the maximum static frictional force:

Maximum static frictional force = coefficient of static friction × normal force

Now, let's calculate the force required to overcome static friction and initiate sliding. This force is equal to the maximum static frictional force:

Force required to overcome static friction = maximum static frictional force

Since the snowboarder is on an incline, this force can be decomposed into two components: one parallel to the incline and one perpendicular to the incline. The component parallel to the incline is responsible for initiating sliding. It can be calculated using:

Force parallel to the incline = force required to overcome static friction × sin(θ)

Where θ is the angle of the incline.

Now, we will compare this force (force parallel to the incline) with the force of gravity acting down the incline (component of force of gravity perpendicular to the incline):

Force perpendicular to the incline = force of gravity × cos(θ)

If the force parallel to the incline is greater than the force perpendicular to the incline, the snowboarder will start sliding.

Therefore, we need to find the angle θ such that:

Force parallel to the incline = Force perpendicular to the incline

Now, you can solve the equation to find the angle θ. Once you have the angle, make sure to answer in units of degrees!

Remember, don't go downhill too fast or you might snowboard past all the humor. Good luck!

To find the angle θ, we need to use the formula for static friction:

Fs = μs * N

where Fs is the force of static friction and N is the normal force.

The normal force N can be calculated using the weight (mg) of the snowboarder, inclined at an angle θ:

N = mg * cos(θ)

Now, the force of static friction can also be written in terms of the weight:

Fs = μs * mg * cos(θ)

In order for the snowboarder to start sliding, the force of static friction needs to be equal to the force component parallel to the incline:

Fs = m * g * sin(θ)

Equating them:

μs * mg * cos(θ) = mg * sin(θ)

Divide both sides by mg:

μs * cos(θ) = sin(θ)

Now, solve for θ:

μs / cos(θ) = tan(θ)

Using the inverse tangent function:

θ = atan(μs / cos(θ))

Substituting the given values μs = 0.55 and μk = 0.4675, we can solve for θ.

Let's calculate θ:

To determine the angle θ the incline must exceed for the snowboarder to start sliding, we can use the concept of static and kinetic friction:

1. Begin by finding the maximum force of static friction (F_s) that can act on the snowboarder. The maximum force of static friction is given by the equation F_s = μ_s * N, where μ_s is the coefficient of static friction and N is the normal force acting on the snowboarder.

2. The normal force (N) can be calculated as N = m * g, where m is the mass of the snowboarder and g is the acceleration due to gravity.

3. Once you have determined the maximum force of static friction (F_s), you can calculate the force component acting parallel to the incline (F_parallel) by using the equation F_parallel = F_s * sin(θ).

4. The force component acting perpendicular to the incline (F_perpendicular) can be calculated as F_perpendicular = F_s * cos(θ).

5. For the snowboarder to start sliding, the force component parallel to the incline (F_parallel) must be greater than the force of kinetic friction (F_k). The force of kinetic friction (F_k) is given by F_k = μ_k * N.

6. Set up the inequality F_parallel > F_k and substitute the relevant equations. Simplify and solve for the angle θ.

7. Once you find the value of θ, convert it to degrees to obtain the final answer.

Remember to substitute the given values for the mass (m), the coefficient of static friction (μ_s), the coefficient of kinetic friction (μ_k), and the acceleration due to gravity (g).

M*g = 60 * 9.8 = 588 N.

Fp = 588*sin A. = Force parallel to the incline.

Fn = 588*Cos A = Normal force.

Fs = us*Fn = 0.55*(588*Cos A = 323.4*Cos A.

Fp-Fs = M*a.
588*sin A-323.4*Cos A = 60*0 = 0,
588*sin A = 323.4*Cos A,
Divide by 588:
sin A = 0.55*Cos A,
Divide by Cos A:
sin A/Cos A = 0.55,
sin A/Cos A = Tan A,
Tan A = 0.55, A = 28.8 Degrees.