A 50 µC charged sphere of mass 0.050kg is stationary and 2.0 m away from a -30µC charged sphere. The 50µC charged sphere released and allowed to approach the other. When it is 1.2 m away, what is the speed of this sphere??
To find the speed of the 50 µC charged sphere when it is 1.2 m away from the -30 µC charged sphere, we can use the concepts of electric potential energy and conservation of energy.
Step 1: Calculate the electric potential energy at a distance of 2.0 m.
The electric potential energy (PE) between two charged spheres is given by the equation:
PE = (k * q1 * q2) / r
Where k is the Coulomb's constant (k = 9 x 10^9 Nm^2/C^2), q1 and q2 are the charges of the two spheres, and r is the distance between them.
In this case, q1 = 50 µC = 50 x 10^-6 C, q2 = -30 µC = -30 x 10^-6 C, and r = 2.0 m. Plugging these values into the equation:
PE1 = (9 x 10^9 * 50 x 10^-6 * (-30 x 10^-6)) / 2.0
Step 2: Calculate the electric potential energy at a distance of 1.2 m.
Using the same equation, plugging in the new distance:
PE2 = (9 x 10^9 * 50 x 10^-6 * (-30 x 10^-6)) / 1.2
Step 3: Apply the principle of conservation of energy.
According to the conservation of energy, the sum of the initial kinetic energy (K.E.1) and the initial potential energy (PE1) is equal to the sum of the final kinetic energy (K.E.2) and the final potential energy (PE2).
K.E.1 + PE1 = K.E.2 + PE2
Since the sphere starts from rest, K.E.1 = 0. Therefore:
0 + PE1 = K.E.2 + PE2
Simplifying the equation:
PE1 - PE2 = K.E.2
Step 4: Calculate the final kinetic energy (K.E.2).
The final kinetic energy can be calculated using the equation:
K.E.2 = (1/2) * m * v^2
Where m is the mass of the sphere and v is its velocity.
In this case, the mass of the sphere is 0.050 kg, and we want to find v.
PE1 - PE2 = (1/2) * m * v^2
Solving for v:
v^2 = (2 * (PE1 - PE2)) / m
Step 5: Calculate the speed (v).
Now that we have the expression for v^2, we can calculate v by taking the square root:
v = √((2 * (PE1 - PE2)) / m)
Plugging in the values we calculated for PE1, PE2, and m:
v = √((2 * (PE1 - PE2)) / 0.050)