An Olympic turkey moving at 20.0 m/s down a 30.0° slope encounters a region of
wet snow and slides 145 m before coming to a halt. What is the coefficient of friction
between the turkey and the snow? The accelerations down the hill is 1.4 m/s^2.
Notes:
1. Fp = Force parallel with the hill.
2. Fn = Normal force.
3. Fk = Force of kinetic friction.
4. Ws = Wt. of skier.
To find the coefficient of friction between the turkey and the snow, we can use the equation:
frictional force = coefficient of friction * normal force
The normal force acting on the turkey can be calculated as:
normal force = mass * gravitational acceleration
where the gravitational acceleration is approximately 9.8 m/s^2.
Let's calculate the turkey's mass first. We can use the equation:
acceleration = (final velocity^2 - initial velocity^2) / (2 * distance)
Given that the acceleration is 1.4 m/s^2, the initial velocity is 20.0 m/s, and the distance is 145 m, we can rearrange the equation to solve for the final velocity:
final velocity = sqrt(2 * acceleration * distance + initial velocity^2)
Plugging in the values, we get:
final velocity = sqrt(2 * 1.4 m/s^2 * 145 m + (20.0 m/s)^2)
= sqrt(406 m^2/s^2 + 400 m^2/s^2)
= sqrt(806 m^2/s^2)
= 28.4 m/s
Now, we can find the mass of the turkey using the equation:
mass = final velocity / (acceleration due to gravity * time)
where time is the time it took for the turkey to come to a halt at the end of the slide. The time can be calculated as:
time = (final velocity - initial velocity) / acceleration
Plugging in the values, we get:
time = (28.4 m/s - 20.0 m/s) / 1.4 m/s^2
= 6 seconds
Now, we can calculate the mass:
mass = 28.4 m/s / (9.8 m/s^2 * 6 s)
≈ 0.486 kg
Finally, we can calculate the coefficient of friction:
coefficient of friction = frictional force / normal force
The frictional force can be calculated using Newton's second law:
frictional force = mass * acceleration down the slope
Plugging in the values, we get:
frictional force = 0.486 kg * 1.4 m/s^2
≈ 0.680 N
And the normal force is:
normal force = mass * gravitational acceleration
= 0.486 kg * 9.8 m/s^2
≈ 4.76 N
Now we can calculate the coefficient of friction:
coefficient of friction = frictional force / normal force
= 0.680 N / 4.76 N
≈ 0.143
Therefore, the coefficient of friction between the turkey and the snow is approximately 0.143.
To find the coefficient of friction between the turkey and the snow, we can use the concept of Newton's second law of motion. The equation is:
F = m * a
where F is the net force, m is the mass of the turkey, and a is the acceleration.
In this case, the net force acting on the turkey is the force due to gravity pulling it downhill (Fg) minus the force due to friction (Ff). The force due to gravity can be calculated as:
Fg = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s²).
The force due to friction can be calculated as:
Ff = μ * N
where μ is the coefficient of friction and N is the normal force.
The normal force can be calculated as:
N = m * g * cos(θ)
where θ is the angle of the slope.
Now, let's break down the problem and calculate the required values step by step:
1. Calculate the force due to gravity:
Fg = m * g
2. Calculate the normal force:
N = m * g * cos(θ)
3. Calculate the force due to friction:
Ff = μ * N
4. Calculate the net force:
F = m * a
5. Set up the equation for the net force:
F = Fg - Ff
6. Solve the equation for the coefficient of friction:
μ = (Fg - F) / N
Now, let's plug in the given values and calculate the coefficient of friction:
Given:
Initial velocity (u) = 20.0 m/s
Angle of the slope (θ) = 30.0°
Distance traveled (s) = 145 m
Acceleration down the hill (a) = 1.4 m/s²
First, we need to calculate the time it took for the turkey to come to a halt. We can use the following kinematic equation:
v^2 = u^2 + 2as
where v is the final velocity and s is the distance.
Rearranging the equation, we get:
v^2 = u^2 + 2as
0 = (20.0 m/s)^2 + 2 * 1.4 m/s^2 * s
s = (0 - (20.0 m/s)^2) / (2 * 1.4 m/s^2)
s = -400 m²/s² / 2.8 m/s²
s = -142.86 m
Since distance cannot be negative, we discard the negative sign. Therefore, the turkey traveled 142.86 m before coming to a halt.
Now, let's calculate the coefficient of friction:
1. Calculate the force due to gravity:
Fg = m * g
Fg = m * 9.8 m/s²
2. Calculate the normal force:
N = m * g * cos(θ)
N = m * 9.8 m/s² * cos(30.0°)
3. Calculate the force due to friction:
Ff = μ * N
4. Calculate the net force:
F = m * a
a = F / m
5. Set up the equation for the net force:
F = Fg - Ff
a = (Fg - Ff) / m
6. Solve the equation for the coefficient of friction:
μ = (Fg - F) / N
By plugging in the given values and calculated values, we can find the coefficient of friction.
Ws = M*g = 9.8M.
Fp = 9.8M*sin30 = 4.9M.
Fn = 9.8M*Cos30 = 8.49M.
Fk = u*Fn = u*8.49M.
Fp-Fk = M*a.
4.9M-u8.49M = M*(-1.4)
Divide by M:
4.9-8.49u = -1.4
-8.49u = -1.4-4.9 = -6.3
u = 0.74.