Two small pith balls, each of mass m = 10.1 g, are suspended from the ceiling of the physics lab by 0.8 m long fine strings and are not moving. If the angle which each string makes with the vertical is è = 42.1°, and the charges on the two balls are equal, what is the magnitude of that charge (in µC)?
If the charged ball is suspended be the string which is deflected by the angle α, the forces acting on it are: mg (downwards), tension T (along the string - to the pivot point), and F (electric force –along the line connecting the charges).
Projections on the horizontal and vertical axes are:
x: T•sin α = F, ….(1)
y: T•cosα = mg. ….(2)
Divide (1) by (2):
T•sin α/ T•cosα = F/mg,
tan α = F/mg.
Since
q1=q2=q.
r=2•L•sinα,
k=9•10^9 N•m²/C²
F =k•q1•q2/r² = k•q²/(2•L•sinα)².
tan α = F/mg =
= k•q²/(2•L•sinα)² •mg.
q = (2•L•sinα) • sqrt(m•g•tanα/k)=
=(2•0.8•sin42.1) •sqrt(0.0101•9.8•tan42.1/9•10^9) =...
To find the magnitude of the charge on the pith balls, you can use Coulomb's law which states that the electrostatic force between two charged objects is proportional to the product of their charges and inversely proportional to the square of the distance between them.
Step 1: Calculate the gravitational force acting on each pith ball.
The gravitational force (F_gravity) acting on each pith ball can be calculated using the formula F_gravity = m * g, where m is the mass of the ball and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the mass of each pith ball is 10.1 g (or 0.0101 kg), we can calculate F_gravity.
F_gravity = 0.0101 kg * 9.8 m/s^2
Step 2: Calculate the tension force in the strings.
The tension force (F_tension) in the strings can be calculated using the formula F_tension = F_gravity / sin(è), where è is the angle between the string and the vertical direction (given as 42.1°).
Let's convert the angle from degrees to radians.
Angle in radians = 42.1° * (π/180)
F_tension = F_gravity / sin(angle in radians)
Step 3: Calculate the electrostatic force between the pith balls.
The electrostatic force (F_electrostatic) between the pith balls is equal to the tension force in the strings, as the balls are in equilibrium.
F_electrostatic = F_tension
Step 4: Calculate the charge on the pith balls.
Using Coulomb's law, we know that F_electrostatic is proportional to the product of the charges (q * q) on the pith balls, so we can write:
F_electrostatic = k * (q * q) / r^2
Where k is the electrostatic constant and r is the distance between the balls (given as the length of the string, which is 0.8 m).
Since the charges on the two balls are equal (q1 = q2 = q), we can rewrite the equation:
F_electrostatic = 2 * k * (q * q) / r^2
Now, equating F_electrostatic to F_tension, we have:
F_tension = 2 * k * (q * q) / r^2
Step 5: Solve for the charge (q).
Rearranging the equation gives us:
q * q = (F_tension * r^2) / (2 * k)
Now, we can substitute the values:
q * q = (F_tension * (0.8 m)^2) / (2 * k)
To find q, we take the square root of both sides of the equation:
q = √[(F_tension * (0.8 m)^2) / (2 * k)]
Finally, calculate the magnitude of the charge by converting from Coulombs to microCoulombs (µC):
Magnitude of charge = q * 10^6 µC
Note: k is the electrostatic constant and has a value of approximately 8.99 * 10^9 N*m^2/C^2.