A cylindrical rod( length L, radius R and density d) is dipped vertically in to a liquid. the rod is connected by a wire to a balance that measures the force to lift the rod. the contact angle
between rod and liquid is θ. If rod is partially immersed so 0.5L is above the surface of liquid. Find the energy needed to raise the rod by small vertical displacement dz. what is the force required for that displacement??
To calculate the energy needed to raise the rod by a small vertical displacement (dz), we need to consider two components: the potential energy of the raised part of the rod and the work done against the buoyant force.
1. Potential Energy:
The potential energy (U) of the raised part of the rod is given by the equation:
U = m * g * h
where m is the mass of the raised part of the rod, g is the acceleration due to gravity, and h is the vertical height of the raised part.
To find the mass of the raised part (m), we need to find the volume of that part and multiply it by the density (d) of the rod. The volume (V) of the raised part is given by the equation:
V = π * R^2 * dz
where R is the radius of the rod and dz is the small vertical displacement.
So, the mass (m) of the raised part is:
m = V * d = π * R^2 * dz * d
Now, substitute the value of m into the potential energy equation:
U = (π * R^2 * dz * d) * g * h
2. Work Done:
The work done is the force (F) applied to lift the rod multiplied by the displacement (dz) over which the force is applied.
From the given information, we know that the rod is connected to a balance that measures the force to lift the rod. So, the force required to lift the rod by dz displacement is equal to the weight of the raised part.
The weight (W) of the raised part is given by the equation:
W = m * g = (π * R^2 * dz * d) * g
Therefore, the work done (Wd) is:
Wd = F * dz = W * dz = (π * R^2 * dz * d) * g * dz
To summarize,
- The energy needed to raise the rod by a small vertical displacement (dz) is U = (π * R^2 * dz * d) * g * h.
- The force required for that displacement is F = (π * R^2 * dz * d) * g.
To calculate the energy needed to raise the rod by a small vertical displacement dz, we can consider the following steps:
Step 1: Determine the volume of the portion of the rod that is submerged in the liquid.
The volume of the submerged portion can be calculated using the formula for the volume of a cylinder:
V_sub = π * R^2 * h_sub
where R is the radius of the rod, and h_sub is the height of the submerged portion.
Given that the length of the rod is L and 0.5L is above the surface of the liquid, the height of the submerged portion is (L - 0.5L) = 0.5L.
Therefore, the volume of the submerged portion is:
V_sub = π * R^2 * 0.5L
Step 2: Calculate the mass of the submerged portion.
The mass of the submerged portion can be calculated by multiplying the volume by the density of the rod:
m_sub = d * V_sub
where d is the density of the rod.
Step 3: Calculate the force required to lift the submerged portion.
The force required to lift the submerged portion is equal to the weight of the rod. The weight is given by:
F_sub = m_sub * g
where g is the acceleration due to gravity.
Step 4: Determine the energy needed to raise the rod by a small vertical displacement dz.
The work done in raising the rod by a small vertical displacement dz is given by the formula:
dW = F_sub * dz
where dW is the work done and dz is the small vertical displacement.
Therefore, the energy needed to raise the rod by a small vertical displacement dz is equal to the work done:
E = dW = F_sub * dz
So, in summary, to determine the energy needed to raise the rod by a small vertical displacement dz, we need to calculate the volume of the submerged portion, then calculate the mass of the submerged portion, and finally, calculate the force required to lift the submerged portion.