In an electricity experiment, a 1.30g plastic ball is suspended on a 56.0cm long string and given an electric charge. A charged rod brought near the ball exerts a horizontal electrical force F⃗ elec on it, causing the ball to swing out to a 25.0∘ angle and remain there.
What is the magnitude of F⃗ elec?
What is the tension in the string?
Well, well, well, looks like we have an electrifying situation here! Let's see if we can lighten the mood by solving this problem with a spark of humor!
To find the magnitude of F⃗ elec, we can use a bit of trigonometry. We've got ourselves a swinging ball, just like Tarzan, but instead of vines, we've got a string and some electric forces. So, let's swing into the calculations!
Given that the ball swings out to a 25.0∘ angle, we can use the sine function to figure out the magnitude of the force. Remember, sine saves the day when it comes to angles and triangles!
sin(25∘) = F⃗ elec / Tension
But wait a second, what in the world is the tension? Fear not, my friend, we'll get to that soon!
Now, for the tension in the string. We can't let things hang loose, can we? To solve this, we need to consider two forces at play: the tension and the gravitational force.
Since the ball is hanging still at an angle, the sum of the horizontal forces is zero. That means the electric force and the horizontal component of the gravitational force balance each other out.
Now, let's bring some balance with the laughter! Why did the scarecrow win an award? Because he was outstanding in his field!
Okay, back to business. The horizontal component of the gravitational force can be calculated as:
F⃗ elec = m * g * sin(25∘)
Where m is the mass of the ball (1.30g) and g is the acceleration due to gravity (approximately 9.8 m/s²). Plug in those values and solve!
And now, drumroll please: To find the magnitude of F⃗ elec, divide the force by the tension value you calculated earlier, using that sin(25∘) value. You got this!
As for the tension in the string, we need to find the vertical component of the gravitational force, which is:
Tension = m * g * cos(25∘)
So there you have it! Two calculations to solve this electrifying problem. Just remember to keep that sense of humor charged up along the way! Good luck!
To find the magnitude of the electrical force (F⃗ elec), we can use the concept of Coulomb's Law. However, since the problem does not provide any charge values, we need to find an alternative solution.
Given that the plastic ball swings out and remains at a 25.0° angle, we can assume that the net force acting on the ball is only the horizontal electrical force (F⃗ elec). The vertical component of the force will be balanced by the tension in the string.
Step 1: Find the horizontal component of the weight force.
Since the ball remains at a tilted angle, we can decompose the weight force into its vertical and horizontal components.
The vertical component can be given as:
F_vertical = mg*cos(θ)
Where,
m = mass of the ball = 1.30 g = 0.00130 kg (converting g to kg)
g = acceleration due to gravity = 9.8 m/s^2 (approx.)
θ = angle of swing = 25.0°
Substituting the values, we find:
F_vertical = (0.00130 kg)(9.8 m/s^2) * cos(25.0°)
Step 2: Find the horizontal component of the force.
The horizontal component is the electrical force (F⃗ elec).
Since the ball is in equilibrium, the horizontal electrical force must be equal to the horizontal component of the weight force.
F_horizontal = F⃗ elec = F_vertical
Substituting the value of F_vertical from step 1, we have:
F⃗ elec = (0.00130 kg)(9.8 m/s^2) * cos(25.0°)
Now we can calculate the magnitude of F⃗ elec.
Step 3: Calculate the magnitude of F⃗ elec.
F⃗ elec = (0.00130 kg)(9.8 m/s^2) * cos(25.0°)
Finally, using a calculator, we can find the magnitude of F⃗ elec, which is equal to evaluating the above expression.
For the tension in the string:
Since the ball remains at a tilted angle and is in equilibrium, the tension in the string will be equal to the vertical component of the weight force.
Step 4: Find the vertical component of the weight force.
F_vertical = mg*sin(θ)
Substituting the values, we find:
F_vertical = (0.00130 kg)(9.8 m/s^2) * sin(25.0°)
Now we can calculate the tension in the string.
Step 5: Calculate the tension in the string.
The tension in the string will be equal to the vertical component of the weight force. So,
Tension = F_vertical = (0.00130 kg)(9.8 m/s^2) * sin(25.0°)
Evaluate the expression using a calculator to find the tension in the string.
To find the magnitude of the electric force, F_elec, we can use the known information about the angle and the length of the string.
The weight of the plastic ball, F_weight, can be calculated using the formula F_weight = mass × gravitational acceleration. The gravitational acceleration is typically 9.8 m/s^2.
F_weight = (0.0013 kg) × (9.8 m/s^2) = 0.01274 N.
The tension in the string, Tension, can be decomposed into vertical and horizontal components. The vertical component is equal to the weight of the ball, while the horizontal component is equal to the electric force.
Since the ball remains at a 25.0° angle, we can calculate the vertical component of tension using the equation Tension_vertical = Tension × sin(angle).
The vertical component of tension is equal to the weight of the ball, so Tension_vertical = F_weight = 0.01274 N.
Now, using the equation Tension_horizontal = Tension × cos(angle), we can find the horizontal component of tension.
Tension_horizontal = (Tension)(cos(angle)).
The total tension can be calculated using the Pythagorean theorem: Tension = sqrt(Tension_vertical^2 + Tension_horizontal^2).
Now let's plug in the values:
Tension = sqrt((0.01274 N)^2 + (Tension × cos(25.0°))^2).
Simplifying the equation:
Tension = sqrt(0.0001625476 N^2 + (Tension × 0.906307787) ^ 2).
Taking the square of both sides:
Tension^2 = 0.0001625476 N^2 + (Tension × 0.906307787)^2.
Rearranging the equation to isolate Tension:
Tension^2 - (Tension × 0.906307787)^2 = 0.0001625476 N^2.
Factoring out Tension^2:
Tension^2 (1 - 0.906307787^2) = 0.0001625476 N^2.
Dividing both sides by (1 - 0.906307787^2):
Tension^2 = 0.0001625476 N^2 / (1 - 0.906307787^2).
Taking the square root of both sides:
Tension = sqrt(0.0001625476 N^2 / (1 - 0.906307787^2)).
After performing the calculations, we get:
Tension ≈ 0.014894 N.
Now that we have the tension in the string, we can find the magnitude of the electric force, F_elec, by using the equation F_elec = Tension_horizontal.
Therefore, the magnitude of F_elec is approximately 0.014894 N.
DRAW THE DIAGRAM
weight of ball: .0013kg*9.8N/kg
horizonal force=weight ball * tangent25
Felce=horizonal force
Tension: weightball/sin25