A mass of 100g is attached to one end of a massless rod, 10cm in length, which is pivoted about the opposite end. The rod is held vertical, with the mass at the top, and released. The rod swings.
A. What is the speed of the mass at the instant that the rod is horizontal?
B. What is the speed of the mass when it is making an angle of 30 degrees below the horizon?
To determine the speed of the mass at different positions, we will need to apply principles of conservation of energy and angular displacement.
A. For the first question, we need to find the speed of the mass when the rod is horizontal. At this position, the potential energy is completely converted into kinetic energy. The potential energy can be calculated using the formula mgh, where m is the mass (100g or 0.1kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height of the mass (10cm or 0.1m). Therefore, the potential energy is:
Potential Energy (PE) = mgh = 0.1kg * 9.8 m/s² * 0.1m = 0.098 J
Since there is no friction or other losses in a massless rod, this potential energy is converted entirely into kinetic energy when the mass is at the horizontal position. The formula for kinetic energy is given by KE = 1/2 * mv², where m is the mass and v is the velocity.
Equating the potential energy to kinetic energy, we have:
PE = KE
0.098 J = 1/2 * 0.1kg * v²
v² = (2 * 0.098 J) / 0.1kg
v² = 1.96 m²/s²
v = √(1.96) = 1.4 m/s
Therefore, the speed of the mass at the instant the rod is horizontal is 1.4 m/s.
B. For the second question, we need to determine the speed of the mass when it is making an angle of 30 degrees below the horizontal. At this position, the mass has both kinetic energy and potential energy. The total mechanical energy (E) is the sum of potential energy and kinetic energy and remains constant throughout the motion.
Let's assume the speed of the mass at this position is v₁.
The potential energy at this position is given by mgh, where h is the vertical displacement of the mass from its highest position to the current position. Here, h can be calculated using trigonometry:
h = 10cm * sin(30°) = 10cm * 0.5 = 5cm = 0.05m
The potential energy (PE) is then:
PE = mgh = 0.1kg * 9.8 m/s² * 0.05m = 0.049 J
To find the kinetic energy (KE), we subtract the potential energy from the total mechanical energy (E):
KE = E - PE
Since E is conserved, it is equal to the initial potential energy:
E = mgh = 0.049 J
Now, let's equate KE with the formula 1/2 * mv₁²:
1/2 * 0.1kg * v₁² = 0.049 J
v₁² = (0.049 J * 2) / 0.1kg
v₁² = 0.98 m²/s²
v₁ = √(0.98) = 1 m/s
Therefore, the speed of the mass when it is making an angle of 30 degrees below the horizontal is 1 m/s.