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 determine the starting height of Tarzan on the tree trunk, we can analyze the motion of Tarzan as he slides down the trunk, moves across the mud-covered platform, swings on the vine, and then releases to land on the other side of the stream.
a) First, let's analyze the motion of Tarzan on the tree trunk:
1. Sliding down the trunk:
Since the tree trunk is hollowed and moist, there is no friction acting on Tarzan as he slides down. Therefore, the only force acting on Tarzan is his weight (mg). The distance Tarzan slides down depends on his initial height on the trunk.
Using the equation for gravitational potential energy, we can equate the initial potential energy to the final kinetic 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), h is the initial height on the tree trunk, and v is the final velocity of Tarzan.
2. Moving across the mud-covered platform:
In this part of the motion, Tarzan moves with constant velocity due to the force of kinetic friction. The force of kinetic friction can be calculated using the equation:
f_friction = μk * mg
Where μk is the coefficient of kinetic friction (0.3 in this case), m is Tarzan's mass (60 kg), and g is the acceleration due to gravity (9.8 m/s^2). The force of friction opposes Tarzan's motion and is equal to the force applied by Tarzan towards the edge of the stream.
3. Swinging on the vine:
As Tarzan reaches the edge of the stream, he grabs onto the vine and swings across. Here, we can consider the conservation of mechanical energy. Tarzan's kinetic energy at the bottom of the swing is converted into potential energy at the highest point of the swing.
Using the conservation of mechanical energy, we have:
(1/2)(m)(v^2) = (m)(g)(h)
Where m is Tarzan's mass (60 kg), v is the velocity of Tarzan at the bottom of the swing (which can be determined by solving the equations of motion), g is the acceleration due to gravity (9.8 m/s^2), and h is the height Tarzan needs to attain to clear the stream.
Tarzan needs to release the vine at this height so that he lands dry on the other side of the stream.
By solving these equations simultaneously, you can find the suitable height on the tree trunk for Tarzan to start.
b) To ensure Tarzan doesn't get injured when he lands on the chair attached to the spring, we need the spring to undergo a maximum compression that can absorb Tarzan's kinetic energy without causing harm.
Using Hooke's Law for springs, we have:
F = k * x
Where F is the force exerted by the spring, k is the spring constant, and x is the compression of the spring.
To determine the spring constant (k) needed, we can use the conservation of mechanical energy. The potential energy at the maximum compression of the spring should be equal to Tarzan's kinetic energy just before he lands.
(1/2)kx^2 = (1/2)mv^2
Where m is Tarzan's mass (60 kg), v is the velocity just before landing (which can be determined using the equations of motion), k is the spring constant, and x is the maximum compression of the spring (0.3 m).
Solving this equation will give you the appropriate spring constant (k) needed for the chair to avoid injuring the actor playing Tarzan.