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.
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 mass of the object multiplied by its acceleration (F = ma).
In this case, the force acting on the metal cylinder is the difference between the pressures inside and outside the tube, multiplied by the area of contact between the plunger and the metal cylinder. We can express this as:
F = (P_inside - P_outside) x A
where F is the force, P_inside is the pressure inside the tube, P_outside is the pressure outside the tube, and A is the area of contact between the plunger and the metal cylinder.
The pressure inside the tube is increased by a factor of 2.79, so the new pressure inside the tube (P_inside) is 2.79 times the pressure outside the tube (P_outside), or:
P_inside = 2.79 x P_outside
Next, we need to calculate the area of contact (A). The area of a cylinder can be found using the formula:
A = 2πrh
where r is the radius and h is the height of the cylinder. In this case, the radius of the metal cylinder is given as 7.07 mm, or 0.00707 meters.
Now, we can calculate the force acting on the metal cylinder:
F = (2.79 x P_outside) x (2π x 0.00707 m x 0.527 kg)
Finally, we can use Newton's second law of motion to find the initial acceleration (a). Rearranging the formula F = ma, we have:
a = F / m
Substituting the values we have calculated, we can find the initial acceleration of the metal cylinder.