You are a member of an alpine rescue team and must project a box of supplies, with mass m, up an incline of constant slope angle α so that it reaches a stranded skier who is a vertical distance h above the bottom of the incline. The incline is slippery, but there is some friction present, with kinetic friction coefficient μk.
Part A
Use the work-energy theorem to calculate the minimum speed v that you must give the box at the bottom of the incline so that it will reach the skier.
Express your answer in terms of some or all of the variables m, g, h, μk, and α.
To calculate the minimum speed v required for the box to reach the skier, we can use the work-energy theorem.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, we need to calculate the work done to lift the box to the skier's height.
First, let's define the variables:
- m: mass of the box
- g: acceleration due to gravity (9.8 m/s^2)
- h: vertical distance between the skier and the bottom of the incline
- μk: kinetic friction coefficient
- α: slope angle
The total work done on the box can be divided into two parts: the work done against gravity and the work done against friction.
1. Work done against gravity:
The force required to lift the box against gravity is mg (mass times acceleration due to gravity). The vertical distance over which this force acts is h. Therefore, the work done against gravity is given by:
Work_gravity = m * g * h
2. Work done against friction:
The frictional force opposes the motion of the box up the incline. The frictional force can be calculated as:
Frictional force = μk * Normal force
The normal force is perpendicular to the incline and is equal to mg * cos(α). Therefore, the frictional force is:
Frictional force = μk * mg * cos(α)
The distance over which this force acts is the horizontal distance traveled along the incline, which can be calculated as:
Distance = h * sin(α)
Therefore, the work done against friction is:
Work_friction = Frictional force * Distance
Work_friction = μk * mg * cos(α) * h * sin(α)
Now, let's apply the work-energy theorem:
Total work done = Work_gravity + Work_friction
Change in kinetic energy = 0, as we want the box to come to rest at the skier's location.
Therefore, the equation becomes:
Work_gravity + Work_friction = 0
Substituting the values for work done against gravity and work done against friction:
m * g * h + μk * mg * cos(α) * h * sin(α) = 0
Now, solve the equation for v:
m * g * h + μk * mg * cos(α) * h * sin(α) = 0
v^2 / 2 - 0 = m * g * h + μk * mg * cos(α) * h * sin(α)
Simplifying the equation, we get:
v^2 = 2 * (m * g * h + μk * mg * cos(α) * h * sin(α))
v = √[2 * (m * g * h + μk * mg * cos(α) * h * sin(α))]
Hence, the minimum speed v required for the box to reach the skier is given by √[2 * (m * g * h + μk * mg * cos(α) * h * sin(α))].
To calculate the minimum speed v required for the box to reach the skier, we need to consider the work done against friction and the gravitational potential energy:
1. Work done against friction:
The work done against friction is given by the equation:
W_friction = μk * N * d
where
- μk is the coefficient of kinetic friction,
- N is the normal force (equal to m * g, where g is the acceleration due to gravity),
- d is the distance traveled along the incline.
Since the box starts from rest, the distance traveled (d) can be calculated using the vertical distance h and the slope angle α:
d = h / sin(α)
Substituting the values in the equation for work done against friction:
W_friction = μk * m * g * h / sin(α)
2. Gravitational potential energy:
The gravitational potential energy is given by the equation:
PE = m * g * h
According to the work-energy theorem, the work done on an object is equal to the change in its mechanical energy. In this case, the work done to move the box up the incline is equal to the change in its mechanical energy.
So, the work done against friction must be equal to the change in gravitational potential energy:
W_friction = -ΔPE
Setting the two equations equal and solving for the minimum speed v:
μk * m * g * h / sin(α) = -m * g * h
Simplifying and solving for v:
v = sqrt(2 * g * h * sin(α) / μk)
Therefore, the minimum speed v required for the box to reach the skier is given by the equation:
v = sqrt(2 * g * h * sin(α) / μk)
KE=PE+W(fr)
KE=mv²/2
PE= mgh
W(fr) =F(fr) •s=µmgcosα•h/sinα
mv²/2 =mgh+µmgcosα•h/sinα
Solve for ‘v’