An axle of a mass of 10 kg, length 10 cm and radius .1 m is free to rotate about the axis which runs the length of the axle, throuhh it's center. A chain of mass 4kg is fastened to the axle at one end, wound exactly 6 times around the axle, and a rock of mass 1.8 kg attached to the open end of the chain. If the axle is released, and ignoring friction, what is the rotational speed of the axle when the entire chain first becomes vertical?
A 1.32 kg sample of some material increases in temperature from 13.5°C to 39.2°C when 1.98×104 J of heat energy is transferred to it. What is the specific heat of the material?
To find the rotational speed of the axle when the entire chain first becomes vertical, we can use the principle of conservation of angular momentum.
First, let's calculate the initial angular momentum of the system before the chain becomes vertical.
The angular momentum of an object is given by the equation:
L = Iω
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia of the axle can be calculated using the equation:
I = (1/2) * m * r^2
where m is the mass of the axle and r is its radius.
Substituting the given values, we have:
I_axle = (1/2) * 10 kg * (0.1 m)^2
= 0.05 kg*m^2
The moment of inertia of the chain and the rock can be approximated by considering the chain as a thin rod rotating about its end. The moment of inertia of a thin rod rotating about one end is given by the equation:
I_rod = (1/3) * m * L^2
where m is the mass of the rod (chain), and L is its length.
The mass of the chain is given as 4 kg, and the length of the chain wound around the axle (rod) is 6 times the length of the axle, which is 10 cm or 0.1 m. So we have:
I_chain = (1/3) * 4 kg * (6 * 0.1 m)^2
= 0.48 kg*m^2
Now, let's consider the entire system. Initially, the chain and the rock are wound around the axle and the system is at rest. The initial angular momentum is zero, as there is no initial rotary motion.
When the entire chain becomes vertical, the chain and the rock are at the bottom. At this point, the entire angular momentum of the system will be due to the rotation of the axle.
The final angular momentum, L_final, is given by the equation:
L_final = I_axle * ω_final
where ω_final is the final angular velocity of the axle.
Since angular momentum is conserved, we can set L_final equal to the initial angular momentum:
L_initial = L_final
0 = I_axle * ω_final
Solving for ω_final:
ω_final = 0 / I_axle
= 0 rad/s
Therefore, when the entire chain first becomes vertical, the rotational speed of the axle is 0 rad/s.