Sinking Object. A stone has a mass of 1.0kg and density 5000kg/m³ is dropped in water. It crossed the surface of the water with an initial velocity of V₀= -0.55m/s ĵ.
a) Calculate the acceleration of the stone
b) Calculate the position of the stone after 1.0 seconds. Use an upward y-axis with its origin at the surface of the water
c) The stone settles on the bottom of the container. What is Fɴ?
g = acceleration of gravity, about 9.81 m/s^2
Vstone = mass stone/density stone
= 1 kg/ 5000 kg/m^3 = 0.2 *10^-3 = 2 * 10^-4 m^3
weight stone =1 g
Density of water = 1000 kg/m^3
Force of buoyancy up on stone = weight of water displaced
= 1,000 * g * Vstone = 10^3 * g * 2 * 10^-4 = 0.2 g
net force down = weight - buoyancy
= 1 g - .2 g = 0.8 g
F = m a
a = F/m = 0.8 g / 1
a = .8 g downward (which we could have guessed from the beginning because the rock is five times the mass of water it displaces)
(b) x = -(1/2)(.8 g) t^2
(c) force up on rock from bottom = weight - buoyancy
= .8 m g
a) To calculate the acceleration of the stone, we can use Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration (a).
The force acting on the object in this case is the buoyant force (Fb), which can be calculated using Archimedes' principle. Archimedes' principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
The weight of the fluid displaced by the stone can be calculated using the formula: weight = density × volume × gravitational acceleration.
First, let's calculate the volume of the stone. The density of the stone is given as 5000 kg/m³, and the mass is given as 1.0 kg. We can use the formula: volume = mass / density.
volume = 1.0 kg / 5000 kg/m³
volume = 0.0002 m³
Next, let's calculate the weight of the fluid displaced by the stone. The density of water is 1000 kg/m³, and the gravitational acceleration is approximately 9.8 m/s².
weight = density × volume × gravitational acceleration
weight = 1000 kg/m³ × 0.0002 m³ × 9.8 m/s²
weight = 0.196 N
Since the buoyant force is equal to the weight of the fluid displaced, the buoyant force acting on the stone is also 0.196 N.
Now, we can use Newton's second law to calculate the acceleration.
F = m × a
0.196 N = 1.0 kg × a
Solving for a, we find that the acceleration of the stone is:
a = 0.196 N / 1.0 kg
a = 0.196 m/s²
Therefore, the acceleration of the stone is 0.196 m/s².
b) To calculate the position of the stone after 1.0 second, we can use the equations of motion under constant acceleration.
The equation for displacement (y) as a function of time (t), initial velocity (V₀), and acceleration (a) is:
y = V₀t + (1/2)at²
In this case, the initial velocity (V₀) is given as -0.55 m/s ĵ, the time (t) is 1.0 second, and the acceleration (a) is 0.196 m/s².
Using these values, we can calculate the displacement of the stone after 1.0 second:
y = (-0.55 m/s) × (1.0 s) + (1/2) × (0.196 m/s²) × (1.0 s)²
y = -0.55 m + 0.098 m
y = -0.452 m
The negative sign indicates that the stone has moved downwards from the surface of the water.
Therefore, the position of the stone after 1.0 second is -0.452 m with respect to the origin at the surface of the water.
c) To calculate the normal force (Fɴ) acting on the stone when it settles on the bottom of the container, we need to consider the forces acting on the stone.
When the stone is at rest on the bottom of the container, the buoyant force (Fb) acting upwards is balanced by the weight of the stone (mg) acting downwards, and the normal force (Fɴ) acting upwards is balanced by the gravitational force (mg) acting downwards.
Since the stone is at rest, the acceleration (a) is 0 m/s², and the net force acting on the stone is zero.
Therefore, we have:
Fɴ - mg = 0
Solving for Fɴ, we find that:
Fɴ = mg
The weight of the stone is mg, where m is the mass of the stone and g is the acceleration due to gravity (approximately 9.8 m/s²).
Fɴ = (1.0 kg) × (9.8 m/s²)
Fɴ = 9.8 N
Therefore, the normal force (Fɴ) acting on the stone when it settles on the bottom of the container is 9.8 N.