A crate of soda of 30kg is set at the top of inclined plane of 45degree to accelerate downward along the plane.calculate the magnitude of acceleration of crate and the velocity of the crate if the length of inclined plane is 3m
Well, well, well, looks like the soda crate is ready for a wild ride! Let's crunch some numbers here:
To find the magnitude of acceleration, we need to determine the component of gravity that acts along the inclined plane. That would be the weight of the soda crate, which is 30 kg multiplied by the acceleration due to gravity (which is approximately 9.8 m/s²).
So, F = m * g = 30 kg * 9.8 m/s².
Now it's time to break out some trigonometry skills! Since the inclined plane is at an angle of 45 degrees, the component of gravity acting down the plane can be found using the sine of the angle:
F_parallel = F * sin(45°).
With this information, we can calculate the magnitude of acceleration. Newton's second law tells us that F = m * a, so:
m * a = F_parallel.
Now let's plug in the values:
30 kg * a = (30 kg * 9.8 m/s²) * sin(45°).
Simplifying that equation will give us the magnitude of acceleration, expressed in meters per second squared.
As for the velocity of the crate, we can use the good ol' kinematic equation: v = u + at. Since the crate starts from rest, the initial velocity (u) is zero. We already solved for the acceleration (a), and the length of the inclined plane (s) is given as 3m.
So, v = 0 + a * t. Here we are looking for the final velocity, so we need to find the time it takes for the crate to slide down the inclined plane.
To find that, we can use another kinematic equation: s = ut + (1/2) * a * t². Again, the initial velocity (u) is zero, and the length of the inclined plane (s) is 3m.
Solving for time (t) in this equation will give us the time it takes for the crate to reach the bottom of the inclined plane. We can then plug that value into the first kinematic equation to find the final velocity (v).
But hey, let's not get too serious here! It's all about fun, right? The crate might just want to take its sweet time sliding down that inclined plane, or perhaps it wants to go full speed and break free from the laws of physics altogether! Regardless, let's crunch those numbers and find out.
To calculate the magnitude of acceleration of the crate, we can use the following formula:
a = g * sin(θ)
where:
a = acceleration
g = acceleration due to gravity (9.8 m/s²)
θ = angle of the inclined plane (45 degrees)
Plugging in the values, we get:
a = 9.8 m/s² * sin(45°)
To calculate the sin(45°) value, we first convert the angle from degrees to radians:
45° * π/180 = 0.7854 radians
Then, we can calculate the sin(45°) using a calculator or by using the sine function in Python:
sin(45°) ≈ 0.7071
Now we can substitute the value of sin(45°) into our equation:
a = 9.8 m/s² * 0.7071
a ≈ 6.928 m/s²
Therefore, the magnitude of acceleration of the crate is approximately 6.928 m/s².
To calculate the velocity of the crate, we can use the following formula:
v = √(2 * a * d)
where:
v = velocity
a = acceleration (6.928 m/s², as calculated above)
d = distance along the inclined plane (3 m, as given in the question)
Plugging in the values, we get:
v = √(2 * 6.928 m/s² * 3 m)
v = √(41.568 m²/s²)
v ≈ 6.449 m/s
Therefore, the velocity of the crate is approximately 6.449 m/s.
To calculate the magnitude of acceleration of the crate and the velocity of the crate, we can use the principles of Newton's laws of motion and trigonometry.
1. First, we need to resolve the gravitational force into two components along the inclined plane. The component of the weight acting parallel to the plane (F ||) will cause the acceleration, while the component perpendicular to the plane (F ⊥) will be balanced by the normal force.
F || = mg * sin(θ)
F ⊥ = mg * cos(θ)
Given:
mass of crate (m) = 30 kg
angle of inclined plane (θ) = 45 degrees
2. Calculate the component of the weight acting parallel to the inclined plane:
F || = 30 kg * 9.8 m/s^2 * sin(45°)
F || = 30 kg * 9.8 m/s^2 * 0.707
F || = 207.90 N (rounded to two decimal places)
3. Now, use Newton's second law (F = ma) to find the acceleration (a):
F = ma
207.90 N = 30 kg * a
Solve for a:
a = 207.90 N / 30 kg
a = 6.93 m/s^2 (rounded to two decimal places)
Therefore, the magnitude of acceleration of the crate is 6.93 m/s^2.
4. To find the velocity of the crate, we can use the kinematic equation:
v^2 = u^2 + 2as
Since the crate starts from rest (initial velocity u = 0 m/s), the equation simplifies to:
v^2 = 2as
Given:
length of the inclined plane (s) = 3 m
acceleration (a) = 6.93 m/s^2
Substitute the values into the equation:
v^2 = 2 * 6.93 m/s^2 * 3 m
v^2 = 41.58 m^2/s^2
Take the square root of both sides to find the velocity (v):
v = √(41.58 m^2/s^2)
v = 6.45 m/s (rounded to two decimal places)
Therefore, the velocity of the crate is approximately 6.45 m/s.