A pendulum string has a length l = 4.0 ft. When the pendulum ball is released at 90 degrees to the vertical, it will swing down. A nail is located distance d below the point of suspension of the pendulum. Show that d must be at least 0.6l if the ball is to swing completely around in a circle centered on the nail.
...So far, all I ve been able to guess is that this has something to do with centrifugal force or something. But I can t find the height...and I have no idea how in the slightest.
To understand why the distance d must be at least 0.6 times the length l of the pendulum string, let's break down the problem step by step:
1. At the highest point of the pendulum swing, when the ball is released at 90 degrees to the vertical, the only force acting on the ball is its weight (mg), directed downwards.
2. As the ball swings downward, it experiences a combination of gravitational force (mg) and the tension in the string. At any point during the swing, the tension in the string can be broken down into two components: the horizontal component (T_h) and the vertical component (T_v).
3. The horizontal component (T_h) of the tension provides the centripetal force necessary to keep the ball moving in a circular path. It points towards the center of the circle, which happens to be our nail.
4. At the highest point of the swing, when the ball is released, the centripetal force (T_h) is equal to the gravitational force (mg). Therefore, we can write the equation: T_h = mg.
5. As the ball swings downward, the tension in the string increases due to the additional force required to maintain circular motion. This increase in tension can be calculated using the equation: T_h = mg + m*a_c, where a_c is the centripetal acceleration.
6. Using the equation for centripetal acceleration, a_c = (v^2) / r, where v is the velocity and r is the radius of the circular path. In this case, r is the distance from the nail to the point of suspension of the pendulum.
7. It can be shown that the velocity of the ball at the lowest point of the swing is maximized, and it is given by the equation: v = sqrt(2gh), where h is the height from the lowest point to the point of suspension.
8. Now, consider the minimum height h at which the ball needs to be released from to complete a full circle centered on the nail. To complete a full circle, the centripetal force required (T_h) at the topmost point of the swing should be equal to the tension in the string at the lowest point.
9. Using the equation for centripetal acceleration (a_c) and the equation for velocity at the lowest point (v), we can write the equation: mg = mg + m*a_c. Simplifying, we get: 0 = m*(v^2) / r, which can be further simplified to: v^2 = 0.
10. However, we know that v cannot be zero since the ball needs to move at the lowest point. Therefore, the equation v^2 = 0 has no solution, which means a full circle cannot be completed unless d is at least 0.6l.
In conclusion, to ensure the pendulum ball swings completely around in a circle centered on the nail, the distance d below the point of suspension of the pendulum must be at least 0.6 times the length l.