Accelerate the Fun with Physics!
a. What is the vertical length h of the submerged portion of the cylinder when it is in equilibrium?
1. M/(rho * pi * r^2)
2. L - r
3. pi * r * M * L
4. r * rho
b. What is the y-component of the acceleration ay of the cylinder immediately after it is displaced from the equilibrium position?
1. -g(y/h)
2. g(y/h)
3. -g(y/L)
4. g(y/L)
c. What is the period T of the cylinder's oscillations after it is released and starts simple harmonic motion?
1. 2π√(m/k)
2. 2π(rho * pi * r^2 * L/g)
3. 2π√(L * g/rho)
4. 2π√(πr^2Lρg/h^2)
Answer:
a. The vertical length h of the submerged portion of the cylinder when it is in equilibrium is given by the equation h = M/(rho * pi * r^2). This formula takes into account the mass, radius, and density of the cylinder, ensuring the cylinder remains stable in the liquid pool. b. The y-component of the acceleration ay of the cylinder immediately after it is displaced from the equilibrium position is given by ay = -g(y/h). This equation reflects the gravitational force acting on the cylinder as it moves from its original position. c. The period T of the cylinder's oscillations after it is released and starts simple harmonic motion is calculated using the formula T = 2π√(πr^2Lρg/h^2). This equation considers the mass, radius, length, density, and acceleration due to gravity to determine the duration of the cylinder's oscillations.
In physics, understanding the equilibrium, acceleration, and oscillations of an object like a cylinder can be both challenging and exciting. Let's delve into the details of each part of the question to accelerate the fun with physics! a. When the cylinder is in equilibrium, the vertical length h of the submerged portion is crucial for maintaining stability. The formula h = M/(rho * pi * r^2) takes the mass, radius, and density of the cylinder into account to ensure it remains balanced in the liquid. This equilibrium position is essential for studying the cylinder's behavior in the pool. b. After the cylinder is displaced from its equilibrium position, the y-component of the acceleration ay plays a significant role in determining the cylinder's motion. The equation ay = -g(y/h) reflects the gravitational force acting on the cylinder as it moves, highlighting the importance of understanding acceleration in physics. c. The period T of the cylinder's oscillations reveals the rhythmic pattern of its motion after being released. The formula T = 2π√(πr^2Lρg/h^2) incorporates various factors like mass, radius, length, density, and gravity to calculate the duration of the oscillations accurately. This simple harmonic motion adds an element of excitement to the cylinder's behavior in the liquid pool. By exploring the equilibrium, acceleration, and oscillations of the cylinder, we can accelerate the fun with physics and deepen our understanding of how objects interact with their surroundings. So, let's dive into the world of physics and uncover the mysteries of motion and equilibrium!