a 5.63kg ball hangs from a vertical pole by a 2.4 long string. Assume the ball is moving at 4.33m/s in a horizontal circle with the string remaining taut. Calculate the angle that the string makes with the pole.
To calculate the angle that the string makes with the pole, we can use trigonometry. We know that the ball is moving in a horizontal circle, so the tension in the string provides the centripetal force to keep the ball moving in a circle.
Let's break down the forces acting on the ball:
1. The weight of the ball: It acts vertically downward, and its magnitude is given by the formula W = mg, where m is the mass of the ball (5.63 kg) and g is the acceleration due to gravity (approximately 9.81 m/s²).
2. The tension in the string: It acts along the string, towards the center of the circle. The magnitude of the tension is equal to the centripetal force required to keep the ball moving in a circle. This force can be calculated using the formula F = ma_c, where F is the force, m is the mass of the ball, and a_c is the centripetal acceleration.
The centripetal acceleration can be calculated using the formula a_c = v²/r, where v is the velocity of the ball (4.33 m/s) and r is the radius of the circle (which is equal to the length of the string, 2.4 m).
Now, let's calculate the centripetal force:
F = ma_c = m(v²/r) = (5.63 kg)(4.33 m/s)²/2.4 m
F ≈ 95.19 N
Since the tension and weight of the ball are acting in the same line, we can consider the vertical component of the tension as equal to the weight of the ball.
Now, let's calculate the vertical component of the tension:
T_vertical = mg = (5.63 kg)(9.81 m/s²) ≈ 55.25 N
The angle that the string makes with the pole can be determined using trigonometry. We can use the tangent function:
tan(θ) = T_vertical / T_horizontal
Where θ is the angle we want to find, T_vertical is the vertical component of tension (55.25 N), and T_horizontal is the horizontal component of tension.
To find the horizontal component of tension, we can use the Pythagorean theorem:
T_horizontal = √(T_total² - T_vertical²)
T_total = 95.19 N (as calculated previously)
T_horizontal = √(95.19² - 55.25²)
T_horizontal ≈ 77.87 N
Now, let's calculate the angle:
tan(θ) = 55.25 N / 77.87 N
θ = arctan(55.25 N / 77.87 N)
Using a calculator, we find that θ ≈ 35.17 degrees.
Therefore, the angle that the string makes with the pole is approximately 35.17 degrees.