A 0.527 kg metal cylinder is placed inside the top of a plastic tube, the lower end of which is sealed off by an adjustable plunger, and comes to rest some distance above the plunger. The plastic tube has an inner radius of 7.07 mm, and is frictionless. Neither the plunger nor the metal cylinder allow any air to flow around them. If the plunger is suddenly pushed upwards, increasing the pressure between the plunger and the metal cylinder by a factor of 2.79, what is the initial acceleration of the metal cylinder? Assume the pressure outside of the tube is 1.00 atm.
vbg
To find the initial acceleration of the metal cylinder, we can use Newton's second law of motion, which states that the force acting on an object is equal to the product of its mass and acceleration. In this case, the force is due to the pressure difference between the plunger and the metal cylinder.
The pressure difference can be calculated using the formula:
ΔP = P1 - P2
where ΔP is the pressure difference, P1 is the pressure applied by the plunger, and P2 is the pressure outside the tube.
In this problem, the pressure outside the tube is given as 1.00 atm, and the pressure between the plunger and the metal cylinder is increased by a factor of 2.79. Therefore, the pressure applied by the plunger is:
P1 = 2.79 * 1.00 atm = 2.79 atm
Now, we can calculate the force acting on the metal cylinder:
F = ΔP * A
where F is the force, ΔP is the pressure difference, and A is the area over which the force is applied.
In this case, the force is acting on the base of the metal cylinder, and the area over which it is applied is the cross-sectional area of the cylinder. The cross-sectional area of a cylinder is given by:
A = π * r^2
where A is the area, π is a mathematical constant (approximately 3.14), and r is the radius of the cylinder.
The radius of the cylinder can be found by dividing the inner radius of the plastic tube by 2:
r = 7.07 mm / 2 = 3.54 mm = 0.00354 m
Now we can calculate the area:
A = π * (0.00354 m)^2 = 3.14 * (0.00354 m)^2 = 3.14 * 1.2536e-5 m^2 = 3.938 mm^2 = 3.938e-6 m^2
Finally, we can calculate the force:
F = 2.79 atm * 3.938e-6 m^2 = 1.097e-5 N
Now we have the force acting on the metal cylinder. Since the force is equal to the product of mass and acceleration, we can rearrange the formula to solve for acceleration:
F = m * a
a = F / m
where a is the acceleration and m is the mass.
The mass of the metal cylinder is given as 0.527 kg. Now we can calculate the initial acceleration:
a = 1.097e-5 N / 0.527 kg = 2.08e-5 m/s^2
Therefore, the initial acceleration of the metal cylinder is 2.08e-5 m/s^2.