A 20 cm long, 180g rod is pivoted at one end. A 17g ball of clay is stuck on the other end. What is the period if the rod and clay swing as a pendulum?
To find the period of a pendulum, we can use the equation:
T = 2π * √(L / g)
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
Let's calculate the length of the pendulum first. Since the rod is 20 cm long, its length will be the distance from the pivot to the center of mass of the clay ball. The clay ball is stuck on the other end of the rod, so the center of mass of the rod and clay system is at the clay ball.
Given that the rod weighs 180g and the clay ball weighs 17g, the center of mass of the system will be closer to the clay ball, but still within the rod.
We can calculate the distance from the pivot to the center of mass using the concept of the center of mass formula:
L = (m1 * x1 + m2 * x2) / (m1 + m2)
where m1 is the mass of the rod, m2 is the mass of the clay ball, x1 is the distance of the rod from the pivot, and x2 is the distance of the clay ball from the pivot.
Let's plug in the values:
m1 = 180g
x1 = 20 cm
m2 = 17g
x2 = 0 cm (since the clay ball is stuck on the other end of the rod)
L = (180g * 20 cm + 17g * 0 cm) / (180g + 17g)
L = (3600 g·cm) / (197 g)
L ≈ 18.27 cm
Now, we can substitute this value of L in the equation for the period:
T = 2π * √(L / g)
T = 2π * √(18.27 cm / g)
To calculate T, we need the value of g. Assuming a standard gravitational acceleration of 9.8 m/s²:
T = 2π * √(0.1827 m / 9.8 m/s²)
Let's calculate T:
T = 2π * √(0.01867 s²/m)
T ≈ 2π * 0.13666 s
T ≈ 0.857 s
Therefore, the period of the pendulum formed by the rod and the clay ball is approximately 0.857 seconds.
To find the period of the pendulum formed by the rod and ball of clay, we can use the formula for the period of a simple pendulum:
T = 2π√(L/g)
Where:
T = period of the pendulum
π = pi (approximately 3.14)
L = length of the pendulum
g = acceleration due to gravity (approximately 9.8 m/s^2)
In this case, the length of the pendulum is the total length of the rod and clay, which is 20 cm (0.2 m). The acceleration due to gravity is always the same, so it's 9.8 m/s^2.
Plugging these values into the formula, we can calculate the period:
T = 2π√(0.2 / 9.8)
T ≈ 2π√(0.0204)
T ≈ 2π * 0.143
T ≈ 0.899 seconds
Therefore, the period of the pendulum formed by the rod and clay is approximately 0.899 seconds.