plzz help me with my homework!!!
Tarzan, played by Patrick Huard, 60 kg, must cross a stream
5 m wide, and full of hungry crocodiles. To do this,
decides to slide along a sloping tree trunk, hollowed
and moist so that there is no friction. After the trunk,
there is a part of 3 m platform covered in mud, with a
coefficient of kinetic friction of 0.3 leading to the edge of
stream. It clings so, without jumping to a vine rigid
8 m with a massless, free to rotate (without friction), in
lower end which is just below where it is
attached (the vine hangs). The vine and he traveled an arc
above the stream. Tarzan must release the vine when
stops momentarily and land, dry, on the other side of
stream!
a) You must ensure that nothing bad will happen to
Tarzan, and for this you must select the tree trunk to be used (starting height of Tarzan).
b) The actor who plays the role of Tarzan wants to land on a chair mounted on a spring so as not to injure,
is stipulated in his contract. This chair is 1 m above the ground and is attached to a spring
can undergo a maximum compression of 0.3 m. What should be the spring constant of the spring to avoid
not damage the player?
To solve this problem, we can break it down into several steps:
a) Selecting the tree trunk:
Let's assume that Tarzan needs to have enough initial height on the tree trunk to safely cross the 5 m wide stream. To calculate this height, we can use the time it takes for Tarzan to slide down the tree trunk and the distance he travels during that time.
First, let's calculate the time it takes for Tarzan to slide down the tree trunk. Since there is no friction, the only force acting on Tarzan is gravity. We can use the kinematic equation:
s = ut + (1/2)at^2,
where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time taken.
In this case, s = 3 m (distance of the mud platform), a = 9.8 m/s^2 (acceleration due to gravity), and u is unknown.
Rearranging the equation to solve for time:
3 = (1/2)(9.8)t^2,
6 = 9.8t^2,
t^2 = 6/9.8,
t = √(6/9.8).
Now, to calculate the initial velocity (u), we can use the equation:
v = u + at,
where v is the final velocity (0 m/s since Tarzan stops momentarily on the vine), and t is the time calculated above.
0 = u + (9.8)(√(6/9.8)),
u = -9.8(√(6/9.8)).
Since velocity cannot be negative in this case, we take only the positive value:
u ≈ 2.49 m/s.
Now that we have the initial velocity, we can use it to calculate the height (h) from which Tarzan should start on the tree trunk. Using the conservation of energy:
mgh = (1/2)mv^2,
where m is Tarzan's mass (60 kg), g is the acceleration due to gravity (9.8 m/s^2), and v is the initial velocity calculated above.
60 x 9.8 x h = (1/2)(60)(2.49)^2,
588h = 37.15,
h ≈ 0.0632 m.
Therefore, Tarzan should start from a height of approximately 0.0632 m on the tree trunk to safely cross the stream.
b) Spring constant for the chair:
To avoid damaging the player, the maximum compression of the spring should be less than or equal to the distance between the chair and the ground (1 m).
Using Hooke's Law:
F = kx,
where F is the force exerted by the spring, k is the spring constant (unknown), and x is the compression of the spring (maximum compression of 0.3 m).
Since the weight of Tarzan (mg) must be equal to the force exerted by the spring, we have:
mg = kx,
(60)(9.8) = k(0.3),
588 = 0.3k,
k ≈ 1960 N/m.
Therefore, the spring constant of the spring should be approximately 1960 N/m to avoid damaging the player.
So, to summarize:
a) Tarzan should start from a height of approximately 0.0632 m on the tree trunk.
b) The spring constant of the spring should be approximately 1960 N/m.
To answer these questions, we need to consider the physics principles involved.
a) First, let's determine the starting height of Tarzan on the tree trunk. Since there is no friction on the tree trunk, we can use the principle of conservation of mechanical energy.
The total mechanical energy of Tarzan when he starts sliding down the tree trunk consists of two components - the potential energy at the starting position and the kinetic energy at the ending position.
The potential energy at the starting position is given by the formula PE = mgh, where m is the mass (60 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the starting height.
The kinetic energy at the ending position is given by the formula KE = 0.5mv^2, where v is the velocity at the ending position.
Since there is no friction, the mechanical energy is conserved, so we can equate the potential energy to the kinetic energy:
mgh = 0.5mv^2
Canceling out the mass and simplifying the equation, we get:
gh = 0.5v^2
Rearranging the equation, we find:
h = 0.5v^2/g
Now, let's consider the vine swing. Since Tarzan stops momentarily at the maximum height of the swing, the kinetic energy here is zero. Using the conservation of mechanical energy again, we can equate the potential energy at the maximum height to the potential energy at the starting position:
mgh = mgh_max
Since the mass cancels out, we have:
gh = gh_max
Substituting the expression for h from before, we get:
0.5v^2/g = gh_max
Simplifying the equation, we find:
h_max = 0.5v^2/g
Therefore, to ensure that nothing bad happens to Tarzan, the starting height on the tree trunk should be equal to the maximum height of the vine swing, which is given by h_max = 0.5v^2/g.
b) Now, let's calculate the spring constant required to ensure that the chair mounted on the spring prevents any damage to the player.
To avoid injury, the maximum force exerted by the spring on the player should be equal to the player's weight.
The force exerted by a spring is given by Hooke's Law as F = kx, where F is the force, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, the maximum displacement x is given as 0.3 m, and the maximum force F should be equal to the weight of the player, which is mg (60 kg * 9.8 m/s^2).
Therefore, we have:
mg = kx
Substituting the given values, we can solve for k:
k = mg/x
k = (60 kg * 9.8 m/s^2) / 0.3 m
k ≈ 1960 N/m
So, the spring constant should be approximately 1960 N/m to avoid damaging the player.