a pendulum bob has by passing through its lowest position. what is its speed when make angle of 60 with the vertical the length of the pendulum is 0.5 .
To find the speed of the pendulum bob when it makes an angle of 60 degrees with the vertical, we can use the principles of conservation of energy and the equation for the speed of an object in circular motion.
Step 1: Find the potential energy (PE) at the highest point of the swing. The potential energy is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. In this case, the bob is at its highest point, so the height h is equal to the length of the pendulum (0.5 m), and the potential energy is given by PE = mgh.
Step 2: Find the potential energy at the lowest point of the swing. At the lowest point, all the potential energy is converted into kinetic energy (KE). Therefore, the potential energy at the highest point is equal to the kinetic energy at the lowest point. So, KE = mgh.
Step 3: Find the speed using the equation for kinetic energy. The equation for kinetic energy is KE = (1/2)mv^2, where m is the mass and v is the velocity. From Step 2, we know that KE = mgh. Therefore, we can equate these two equations to find the velocity: mgh = (1/2)mv^2.
Step 4: Cancel out the mass and simplify the equation. Since the mass appears on both sides of the equation, we can cancel it out. This leaves us with gh = (1/2)v^2.
Step 5: Solve for the velocity. Rearranging the equation, we get v^2 = 2gh. To find the velocity v, we take the square root of both sides: v = √(2gh).
Now, let's substitute the given values into the equation:
g = acceleration due to gravity = 9.8 m/s^2 (standard value)
h = height from the lowest point to the highest point = 0.5 m
v = √(2 * 9.8 m/s^2 * 0.5 m)
v = √(9.8 m^2/s^2)
v ≈ 3.13 m/s
So, the speed of the pendulum bob when it makes an angle of 60 degrees with the vertical is approximately 3.13 m/s.