To find the tension in the string and the charge on the cork ball, we need to analyze the forces acting on the ball when it is in equilibrium. The forces involved are the gravitational force acting downward, the tension in the string, and the electric force acting due to the electric field.
Identifying the Forces
Let's break down the forces acting on the cork ball:
- Gravitational Force (Weight): This acts downward and can be calculated using the formula:
W = mg, where m is the mass and g is the acceleration due to gravity.
- Tension (T): This acts along the string at an angle θ with the vertical.
- Electric Force (F_e): This acts horizontally due to the electric field and can be calculated using the formula:
F_e = qE, where q is the charge and E is the magnitude of the electric field.
Calculating the Gravitational Force
Given that the mass of the cork ball is 1 g (which is 0.001 kg), and using g = 10 m/s², we can calculate the weight:
W = mg = 0.001 \times 10 = 0.01 N
Setting Up the Equilibrium Conditions
In equilibrium, the net force in both the vertical and horizontal directions must be zero. We can resolve the tension into its components:
- The vertical component: T_y = T \cos(θ)
- The horizontal component: T_x = T \sin(θ)
For vertical equilibrium:
T \cos(θ) = W
For horizontal equilibrium:
T \sin(θ) = F_e = qE
Calculating Tension
From the vertical equilibrium equation:
T \cos(37°) = 0.01 N
Using cos(37°) = 0.8 (since sin(37°) = 0.6), we can substitute:
T \cdot 0.8 = 0.01
Solving for T gives:
T = 0.01 / 0.8 = 0.0125 N
Finding the Charge on the Ball
Now, we can use the horizontal equilibrium condition to find the charge:
T \sin(37°) = qE
Substituting the values we have:
0.0125 \cdot 0.6 = q \cdot (3 \hat{i} + 5 \hat{j}) \cdot 10^5
Calculating the left side:
0.0075 = q \cdot (3 \hat{i} + 5 \hat{j}) \cdot 10^5
To find the magnitude of the electric field, we can calculate:
E = \sqrt{(3 \cdot 10^5)^2 + (5 \cdot 10^5)^2} = \sqrt{9 + 25} \cdot 10^5 = \sqrt{34} \cdot 10^5
Now, we can express the charge q in terms of the electric field:
q = 0.0075 / (3 \cdot 10^5) = 2.5 \times 10^{-8} C
Summary of Results
In summary, the tension in the string is 0.0125 N, and the charge on the cork ball is approximately 2.5 x 10^-8 C. This analysis demonstrates how forces interact in an electric field and how equilibrium conditions can be applied to solve for unknown quantities.
