A 2.3 kg ball tied to a string fixed to the ceiling is pulled to one side by a force Farrowbold to an angle of θ = 26.4°.
Just before the ball is released and allowed to swing back and forth, how large is the force Farrowbold that is holding the ball in position?
Just before the ball is released and allowed to swing back and forth, what is the tension in the string?
To determine the force F_arrowbold that is holding the ball in position, we can apply Newton's second law of motion. According to this law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, when the ball is held in position, it is not moving, so its acceleration is zero. Therefore, the net force on the ball is also zero. We can analyze the forces acting on the ball to find the force F_arrowbold.
The forces acting on the ball are:
1. The force of gravity (weight) pulling the ball downward, given by 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).
2. The tension force in the string, which is what we're trying to find (let's call it T_string).
3. The force F_arrowbold, pulling the ball sideways.
Since the ball is held in position, the force F_arrowbold is counteracting the horizontal component of the force of gravity. The horizontal component of the force of gravity can be calculated using trigonometry.
The horizontal force component (F_horizontal) is given by: F_horizontal = F_gravity * sin(theta), where theta is the angle between the downward force of gravity and the horizontal direction.
Now, since the net force is zero, we can write the equation:
Net force = F_arrowbold + F_horizontal = 0.
Therefore, we have:
F_arrowbold = -F_horizontal = - F_gravity * sin(theta).
To find the tension in the string just before the ball is released, we can use the same logic since the net force is still zero. The tension force, T_string, is equal to the vertical component of the force of gravity, which is given by: T_string = F_gravity * cos(theta).
Now, plugging in the values, we can solve for both F_arrowbold and T_string:
- F_arrowbold = - (m * g * sin(theta))
T_string = m * g * cos(theta).
Note: The negative sign for F_arrowbold represents the fact that the force is acting in the opposite direction of the horizontal component of the force of gravity.
Given the mass of the ball (2.3 kg) and the angle (θ = 26.4°), you can substitute these values into the equations above and calculate F_arrowbold and T_string.