An m𝑚 = 1.81-kg ball tied to a string fixed to the ceiling is pulled to one side by a force F at an angle of θ𝜃 = 25.4°. Just before the ball is released and allowed to swing back and forth, how large is the force F that is holding the ball in position?

To solve this problem, we need to understand that the force F holding the ball in position is balanced by the gravitational force acting on the ball. The gravitational force can be calculated using the formula F = m*g, where m is the mass of the ball and g is the acceleration due to gravity (approximately 9.8 m/s^2).

1. Calculate the gravitational force: F_gravity = m * g = 1.81 kg * 9.8 m/s^2 = 17.738 N.

2. The force F is acting at an angle, so only the component of the force that is acting in the same direction as the gravitational force is balancing the gravitational force. This component can be calculated using the formula F_component = F * cos(θ).

3. We can set up the following equation to solve for F: F_gravity = F_component.

4. Substitute the known values into the equation: 17.738 N = F * cos(25.4°).

5. Solve for F: F = 17.738 N / cos(25.4°) = 19.6 N.

So, the force F that is holding the ball in position is approximately 19.6 N.

To solve this problem, we need to understand that the force F holding the ball in position is balanced by the gravitational force acting on the ball. The gravitational force can be calculated using the formula F = m*g, where m is the mass of the ball and g is the acceleration due to gravity.

Here are the steps to solve the problem:

1. Calculate the gravitational force acting on the ball. Given that the mass m of the ball is 1.81 kg and the acceleration due to gravity g is approximately 9.8 m/s², the gravitational force Fg = m*g = 1.81 kg * 9.8 m/s² = 17.738 N.

2. The force F holding the ball in position can be resolved into two components: one vertical (Fv) and one horizontal (Fh). The vertical component Fv balances the gravitational force, and the horizontal component Fh is what causes the ball to swing when released.

3. Since the force F is acting at an angle θ of 25.4° to the vertical, we can use trigonometry to relate F, Fv, and θ. Specifically, cos(θ) = Fv / F.

4. Rearranging the above equation gives F = Fv / cos(θ). Substituting the known values gives F = 17.738 N / cos(25.4°).

5. Using a calculator to compute the cosine of 25.4° and divide 17.738 N by this value gives the magnitude of the force F holding the ball in position.

Remember to check that your calculator is set to degrees before performing the calculation, as the angle θ is given in degrees.