An crate weighing 576 N is resting on a plane inclined 31° above the horizontal.
(a) Calculate the magnitude of the acceleration (ignore friction).
(b) After 4.50 s, how fast will the crate be moving?
To solve part (a) of the problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In this case, the net force is the weight of the crate (576 N), and the mass can be calculated using the formula:
mass = weight / acceleration due to gravity
= 576 N / 9.81 m/s^2
≈ 58.76 kg
Now, we can calculate the magnitude of the acceleration using the component of the weight acting parallel to the inclined plane. The equation for that is:
acceleration = gravity * sin(theta)
= 9.81 m/s^2 * sin(31°)
≈ 5.12 m/s^2
Therefore, the magnitude of the acceleration is approximately 5.12 m/s^2.
For part (b) of the problem, we can use the kinematic equation:
final velocity = initial velocity + (acceleration * time)
Since the crate starts from rest (initial velocity = 0 m/s) and has an acceleration of 5.12 m/s^2, we can substitute those values into the equation along with the given time (4.50 s):
final velocity = 0 m/s + (5.12 m/s^2 * 4.50 s)
= 23.04 m/s
Therefore, after 4.50 s, the crate will be moving with a speed of approximately 23.04 m/s.
To find the answers to these questions, we can use a few principles of physics, such as the force of gravity, the components of force on an inclined plane, and the equations of motion.
(a) To calculate the magnitude of the acceleration, we can use the force of gravity acting on the crate and resolve it into its components. The force of gravity can be found using the formula:
Force of gravity (Fg) = mass (m) * acceleration due to gravity (g)
The acceleration due to gravity is approximately 9.8 m/s². To find the mass of the crate, we can use the formula:
Force of gravity (Fg) = mass (m) * acceleration due to gravity (g)
Given that the weight of the crate is 576 N, we can rearrange the equation to solve for mass:
mass (m) = Force of gravity (Fg) / acceleration due to gravity (g)
m = 576 N / 9.8 m/s²
m ≈ 58.78 kg
Now, let's calculate the component of the weight of the crate acting down the inclined plane. The weight component parallel to the plane can be found using the equation:
Force parallel to the plane (Fp) = Force of gravity (Fg) * sin(angle)
In this case, the angle is 31°. Let's calculate it:
Fp = 576 N * sin(31°)
Fp ≈ 298.98 N
Now, we can calculate the acceleration using the equation:
acceleration (a) = Force parallel to the plane (Fp) / mass (m)
a = 298.98 N / 58.78 kg
a ≈ 5.08 m/s²
So, the magnitude of the acceleration is approximately 5.08 m/s².
(b) To find how fast the crate will be moving after 4.50 s, we can use the equations of motion. Since the crate starts from rest, we can use the equation:
Final velocity (v) = Initial velocity (u) + (acceleration * time)
Given that the initial velocity (u) is 0 m/s, the acceleration (a) is 5.08 m/s², and the time (t) is 4.50 s, we can calculate the final velocity:
v = 0 m/s + (5.08 m/s² * 4.50 s)
v ≈ 22.86 m/s
So, the crate will be moving at approximately 22.86 m/s after 4.50 s.