A skier of mass 70.4 kg, starting from rest, slides down a slope at an angle $\theta$ of 36.8° with the horizontal. The coefficient of kinetic friction is 0.14. What is the net work, i.e. net gain in kinetic energy, (in J) done on the skier in the first 8.4 s of descent?
M*g = 70.4kg * 9.8N/kg = 689.9 N. = Wt.
of skier.
Fp = 689.9*sin36.8 = 413.3 N. = Force
parallel to the incline.
Fn = 689.9*Cos36.8 = 552.4 N. = Normal
force = Force perpendicular to the incline.
Fk = u*Fn = 0.14 * 552.4 = 77.34 N. =
Force of kinetic friction.
a = (Fp-Fk)/M = (413.3-77.34)/70.4 = 4.77 m/s^2.
V = Vo + a*t
V = 0 + 4.77*8.4 = 40.07 m/s
W = KE = M*V^2/2 =
70.4*40.07^2/2 = 56,517 J.
To find the net work done on the skier, we need to calculate the total kinetic energy gained by the skier in the given time interval.
The work done on an object is equal to the change in its kinetic energy. So, we need to find the initial and final kinetic energies of the skier and then subtract them to get the net work.
Given:
Mass of the skier, m = 70.4 kg
Angle of the slope, θ = 36.8°
Coefficient of kinetic friction, μ = 0.14
Time interval, t = 8.4 s
The initial kinetic energy of the skier is zero because the skier starts from rest.
To find the final kinetic energy, we need to find the final velocity of the skier.
The gravitational force, acting in the direction of motion, can be resolved into two components:
- The component parallel to the slope: m * g * sin(θ)
- The component perpendicular to the slope: m * g * cos(θ)
The parallel component provides the acceleration of the skier down the slope. Since there is a frictional force acting opposite to the motion, the equation of motion becomes:
m * a = m * g * sin(θ) - μ * m * g * cos(θ)
Simplifying the equation, we get:
a = g * (sin(θ) - μ * cos(θ))
Now, we can use the equation of motion to find the final velocity of the skier.
v = u + a * t
Since the skier starts from rest (u = 0), the equation becomes:
v = a * t
Plugging in the values, we get:
v = (g * (sin(θ) - μ * cos(θ))) * t
Now, we can calculate the final kinetic energy:
KE_final = (1/2) * m * v^2
Plugging in the values, we get:
KE_final = (1/2) * 70.4 * ((g * (sin(θ) - μ * cos(θ))) * t)^2
Finally, the net work done on the skier is the difference between the final and initial kinetic energies:
Net Work = KE_final - KE_initial
= KE_final - 0
= KE_final
Plugging in the values and calculating the expression will give you the net work done on the skier.
To find the net work done on the skier, we need to calculate the net force acting on the skier and then use the work-energy principle.
First, let's calculate the gravitational force acting on the skier. The gravitational force can be calculated using the equation:
F_gravity = m * g
where m is the mass of the skier and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gravity = (70.4 kg) * (9.8 m/s^2)
F_gravity = 689.12 N
Now, we need to determine the component of the gravitational force acting parallel to the slope. This can be calculated using the equation:
F_parallel = F_gravity * sin(θ)
where θ is the angle of the slope with the horizontal.
F_parallel = 689.12 N * sin(36.8°)
F_parallel = 402.64 N
Next, we need to calculate the force of kinetic friction acting on the skier. The force of kinetic friction can be calculated using the equation:
F_friction = μ * F_normal
where μ is the coefficient of kinetic friction and F_normal is the component of the gravitational force acting perpendicular to the slope.
F_normal = F_gravity * cos(θ)
F_normal = 689.12 N * cos(36.8°)
F_normal = 568.63 N
F_friction = 0.14 * 568.63 N
F_friction = 79.41 N
The net force acting on the skier is given by the difference between the parallel component of the gravitational force and the force of kinetic friction:
F_net = F_parallel - F_friction
F_net = 402.64 N - 79.41 N
F_net = 323.23 N
Now, we can calculate the net work done on the skier using the equation:
Net work = F_net * d * cos(θ)
where d is the distance traveled by the skier in the first 8.4 s.
To find the distance, we can use the equation of motion for uniformly accelerated linear motion:
d = v_i * t + (1/2) * a * t^2
where v_i is the initial velocity (0 m/s in this case), t is the time (8.4 s), and a is the acceleration.
Since the skier starts from rest, the only acceleration acting on the skier is the component of gravity parallel to the slope, which can be calculated using:
a = g * sin(θ)
a = 9.8 m/s^2 * sin(36.8°)
a = 5.9 m/s^2
Now, we can calculate the distance traveled:
d = (0 m/s) * (8.4 s) + (1/2) * (5.9 m/s^2) * (8.4 s)^2
d = 0 + 222.456 m
d = 222.456 m
Finally, we can calculate the net work done:
Net work = (323.23 N) * (222.456 m) * cos(36.8°)
Using a calculator:
Net work ≈ 21491.43 J
Therefore, the net work done on the skier in the first 8.4 s of descent is approximately 21491.43 J.