A 2.8-kg block slides down a frictionless plane that makes an angle of 57° with the horizontal. The acceleration of the block is (g = 10 m/s2):
To find the acceleration of the block sliding down the plane, we first need to resolve the forces acting on the block.
1. Determine the components of the gravitational force acting on the block:
The force of gravity can be split into two components: one parallel to the plane and one perpendicular to the plane.
The component parallel to the plane is given by: m * g * sin(θ), where θ is the angle of the plane.
The component perpendicular to the plane is given by: m * g * cos(θ).
2. The acceleration of the block is caused by the parallel component of the gravitational force. Therefore, the net force in the direction parallel to the plane is: m * g * sin(θ).
3. Finally, we can calculate the acceleration using Newton's second law:
Net force = mass * acceleration
Plugging in the values, we can calculate the acceleration:
m * g * sin(θ) = m * a
Substituting the given values:
2.8 kg * 10 m/s^2 * sin(57°) = 2.8 kg * a
Now, solve for the acceleration:
a = (2.8 kg * 10 m/s^2 * sin(57°)) / 2.8 kg
a = 10 m/s^2 * sin(57°)
Evaluating the expression, we get:
a ≈ 8.57 m/s^2 (rounded to two decimal places)
Therefore, the acceleration of the block sliding down the frictionless plane is approximately 8.57 m/s^2.
To find the acceleration of the block, 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 only force acting on the block is its weight (mg), where m is the mass of the block and g is the acceleration due to gravity.
Let's break down the weight force into its components. The weight force can be divided into two perpendicular directions: one parallel to the incline and one perpendicular to the incline.
The component of the weight force acting along the incline can be calculated using the formula:
Force along incline = Weight * sin(θ)
where θ is the angle that the incline makes with the horizontal.
In this case, the mass of the block is given as 2.8 kg, and the angle of the incline is given as 57°. The acceleration due to gravity is constant and equal to 10 m/s^2.
So, the force along the incline can be calculated as:
Force along incline = (2.8 kg) * (10 m/s^2) * sin(57°)
Now, since there is no friction in the system, the net force acting on the block is equal to the force along the incline. Therefore, the net force is:
Net force = Force along incline = (2.8 kg) * (10 m/s^2) * sin(57°)
Finally, we can use Newton's second law to find the acceleration:
Net force = mass * acceleration
Therefore,
(2.8 kg) * (10 m/s^2) * sin(57°) = (2.8 kg) * acceleration
Simplifying the equation, we get:
Acceleration = (10 m/s^2) * sin(57°)
Using a calculator, we find:
Acceleration ≈ 8.50 m/s^2
Therefore, the acceleration of the block is approximately 8.50 m/s^2.