Two rock climbers, Bill and Karen, use safety ropes of similar length. Karen's rope is more elastic, called a dynamic rope by climbers. Bill has a static rope, not recommended for safety purposes in pro climbing. Karen falls freely about 1.7 m and then the rope stops her over a distance of 1.0 m.
(a) Estimate, assuming that the force is constant, how large a force she will feel from the rope. (Express the result in multiples of her weight.)
____ xKaren's weight
(b) In a similar fall, Bill's rope stretches by 25 cm only. How many times his weight will the rope pull on him?
___ x Bill's weight
Which climber is more likely to be hurt?
a) Bill
b) Both are equally subject to injury.
c) Karen
3 times
In falling freely 1.7 m, both climbers reach a velocity of sqrt(2g*1.7) = 5.77 m/s and a kinetic energy M g *1.7
The average force needed to stop in a distance is given by
F = KE/X = 1.7Mg/X
That is 1.7/1 = 1.7 times the body weight for Karen and 1.7/0.25 = 6.8 times the body weight of Bill
How would you answer the last question?
Where the answer ?
(a) To estimate how large a force Karen will feel from the rope, we can use the concept of work done. The work done is equal to the force applied multiplied by the distance over which the force acts. In this case, the work done by the rope is equal to the force applied (in multiples of Karen's weight) multiplied by the distance the rope stops Karen (1.0 m).
We can calculate the work done by the rope by using the formula:
Work = Force x Distance
Let's assume Karen's weight is W (in Newtons), and we need to find the force in multiples of her weight. So, the force applied can be denoted as F = xW, where x is the factor we need to find.
Using the formula for work, we have:
Work = F x Distance
Since the work done by the rope is equal to the change in kinetic energy (K.E.) of Karen, we can write:
K.E. = Work
Considering that Karen falls freely for 1.7 m before the rope stops her, we can calculate the initial potential energy of Karen as:
Initial potential energy = m x g x h
where m is the mass of Karen, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the fall (1.7 m).
The initial potential energy gets converted into kinetic energy before the rope stops Karen. So, we can write:
K.E. = Initial potential energy = m x g x h
Since the work done by the rope is equal to the change in kinetic energy, we can write:
Work = m x g x h
Now, equating the two expressions for work, we get:
F x Distance = m x g x h
Since the distance over which the rope stops Karen is given as 1.0 m, the equation becomes:
F = (m x g x h) / Distance
We can now substitute the given values into the equation to find the force (F) in multiples of Karen's weight (W).
(b) For Bill's fall, we are given that the rope stretches by 25 cm (0.25 m) only. We need to find how many times his weight the rope pulls on him.
Similar to the previous calculation, we can use the concept of work done to determine the force exerted by the rope on Bill. Using the same formula as before, we have:
Work = Force x Distance
Since the rope stretches by 0.25 m, the work done by the rope is given by:
Work = Force x Distance = F x 0.25
Again, equating the work done by the rope to the change in kinetic energy,
we have:
F x Distance = m x g x h
Substituting the values given, we have:
F x 0.25 = m x g x h
From this equation, we can solve for the force (F) in multiples of Bill's weight (W).
To determine which climber is more likely to be hurt, we need to consider the force exerted on each of them. The climber who experiences a higher force is more likely to be hurt.