A ball of mass 1.76 kg is tied to a string of length 4.19 m as shown in Figure P6.50. The ball is initially hanging vertically and is given an initial velocity of 9.4 m/s in the horizontal direction. The ball then follows a circular arc as determined by the string. What is the speed of the ball when the string makes an angle of 31.5° with the vertical?
To find the speed of the ball when the string makes an angle of 31.5° with the vertical, we first need to understand the physics behind the problem.
When the ball is in motion, it experiences two forces: the gravitational force (weight) pulling it downward, and the tension force of the string pulling it towards the center of the circular arc.
At the beginning, the ball has an initial velocity of 9.4 m/s in the horizontal direction. As it moves along the circular arc, its velocity changes because of the centripetal acceleration.
To solve this problem, we can use the law of conservation of mechanical energy. The mechanical energy, which includes both kinetic energy and potential energy, remains constant throughout the motion.
The initial potential energy of the ball is given by mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height of the ball above some reference point (in this case, we can consider it to be at the bottom of the circle). Since the ball is initially hanging vertically, the height h is zero, and the initial potential energy is also zero.
The initial kinetic energy of the ball is given by ½mv^2, where v is the initial velocity of the ball in the horizontal direction. Plugging in the known values, the initial kinetic energy is ½(1.76)(9.4)^2.
At the final point when the string makes an angle of 31.5° with the vertical, the potential energy and kinetic energy can be calculated using the height and speed of the ball at that point.
The final potential energy is given by mgh, where h is the height of the ball above the reference point. In this case, h can be found using trigonometry: h = l - lcosθ, where l is the length of the string and θ is the angle the string makes with the vertical. Plugging in the known values, the final potential energy is (1.76)(9.8)(4.19 - 4.19cos(31.5°)).
The final kinetic energy can be calculated using the velocity of the ball at that point. The velocity can be found using the formula v = ωr, where ω is the angular velocity and r is the radius of the circular arc. The angular velocity can be found using the formula ω = v/r, where v is the speed of the ball and r is the radius. Plugging in the known values, the angular velocity is (9.4)/(4.19sin(31.5°)). The radius can be found using the formula r = lsinθ, where l is the length of the string and θ is the angle the string makes with the vertical. Plugging in the known values, the radius is (4.19)(sin(31.5°)).
Finally, the final kinetic energy is given by ½mv^2, where v is the speed of the ball at that point. Plugging in the known values, the final kinetic energy is ½(1.76)(v)^2.
Since the total mechanical energy remains constant, we can equate the initial kinetic energy to the final potential energy plus the final kinetic energy:
½(1.76)(9.4)^2 = (1.76)(9.8)(4.19 - 4.19cos(31.5°)) + ½(1.76)(v)^2.
Now, we can solve this equation for the speed of the ball, v. It involves some algebraic manipulations and solving for v. The final value of v will give us the speed of the ball when the string makes an angle of 31.5° with the vertical.