The caber toss is a traditional Scottish sport that involves hurling a caber, which is essentially a large piece of a tree: a caber is a 6 m long, 80 kg tree trunk. Large strong people hurl these and the goal is to get the caber to land as far away as possible and rotate in the air, so what was the highest part of the trunk initially is actually the part that hits the gound first. See this clip to understand how the caber rotates.
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It's HARD to do this. To see how hard, consider a perfectly vertical caber with one end on the ground. You then launch the caber vertically with some speed v and give it a rotation. What is the minimum kinetic energy in Joules you need to give the caber so that when it lands the caber is perfectly vertical again, but the OTHER end of the caber hits the ground?
Details and assumptions:-
The acceleration of gravity is -9.8m/s^2.
The caber can be modeled as a uniform rod.
To find the minimum kinetic energy required to achieve the desired outcome, we need to consider the conservation of energy. Initially, the caber has potential energy due to its height above the ground and no kinetic energy. When the caber lands, it will have kinetic energy due to its motion and no potential energy.
Let's go through the steps to calculate the minimum kinetic energy:
1. Determine the change in potential energy: The potential energy of the caber initially is given by the equation PE_initial = mgh, where m is the mass of the caber, g is the acceleration due to gravity, and h is the height of the caber above the ground. In this case, h is the length of the caber, so h = 6 m.
PE_initial = (80 kg)(9.8 m/s^2)(6 m)
PE_initial = 4704 J
Since the other end of the caber will hit the ground, we want the final potential energy to be zero.
2. Equate initial and final potential energy: The change in potential energy is given by the equation ΔPE = PE_final - PE_initial. In this case, ΔPE = -4704 J.
3. Convert the change in potential energy to kinetic energy: According to the principle of conservation of energy, the change in potential energy is equal to the change in kinetic energy. Thus, ΔKE = -4704 J.
4. Use the formula for kinetic energy: The kinetic energy of a rotating object can be calculated using the equation KE = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
Since the caber is assumed to be a uniform rod, the moment of inertia can be calculated as I = (1/3)ml^2, where m is the mass of the caber and l is its length.
I = (1/3)(80 kg)(6 m)^2
I = 960 kg•m^2
Substituting these values into the formula for kinetic energy:
ΔKE = (1/2)(960 kg•m^2)ω^2
5. Determine the relationship between angular velocity and linear velocity: The linear velocity of a rotating object can be related to its angular velocity by the equation v = rω, where v is the linear velocity and r is the distance from the axis of rotation.
In this case, the distance from the axis of rotation to the center of mass of the caber can be considered as half of its length, so r = 3 m.
Thus, v = 3 mω.
6. Convert the relationship into kinetic energy terms: The kinetic energy in terms of linear velocity is KE = (1/2)mv^2.
Substituting the linear velocity relationship into the kinetic energy equation:
ΔKE = (1/2)m(3 mω)^2
ΔKE = (1/2)(80 kg)(3 m)^2ω^2
ΔKE = 3600 Jω^2
7. Equate the change in kinetic energy to the change in potential energy: Since we want the final potential energy to be zero, the change in kinetic energy is equal to the change in potential energy, ΔKE = -4704 J.
3600 Jω^2 = -4704 J
The negative sign indicates that the direction of the angular velocity is opposite to the initial direction we assumed. However, we are interested only in the magnitude, so we can work with:
3600 ω^2 = 4704
Dividing both sides by 3600:
ω^2 = 4704 / 3600
ω^2 ≈ 1.3067
Taking the square root of both sides:
ω ≈ 1.143 rad/s
8. Calculate the kinetic energy: Finally, we can substitute this value of ω into the equation for kinetic energy to find the minimum kinetic energy required.
KE = (1/2)(960 kg•m^2)(1.143 rad/s)^2
KE ≈ 618.1 J
Therefore, to achieve the desired outcome, the minimum kinetic energy required is approximately 618.1 Joules.