A mass of 10kg is placed on the top of a plane inclined at 45 degrees to the horizontal and allowed to slide down. What is the acceleration of the mass if the frictional force opposing is 3.2N.
mg = 10kg * 9.8 = 98N @ 45deg.
Fp = 98*sin(45) = 69.3N = Force acting parallel to plane and downward.
Fn = Fp - Ff = 69.3 - 3.2 = 66.1N.
a = Fn / m = 66.1 / 10kg = 6.61m/s^2.
To find the acceleration of the mass, we can use the following formula:
F_net = m * a
where F_net is the net force acting on the mass, m is the mass of the object, and a is the acceleration.
In this case, we need to determine the net force acting on the mass.
The forces acting on the mass are:
- Weight (mg), which is the force due to gravity.
- Friction force (F_friction), which is opposing the motion.
First, let's determine the weight of the mass using the formula:
Weight (mg) = mass (m) * acceleration due to gravity (g)
The acceleration due to gravity is approximately 9.8 m/s^2.
Weight (mg) = 10 kg * 9.8 m/s^2 = 98 N
Since the plane is inclined at 45 degrees to the horizontal, we can determine the component of weight acting along the incline.
Component of weight along the incline (mg sinθ) = Weight (mg) * sin(θ)
θ = 45 degrees
Component of weight along the incline (mg sinθ) = 98 N * sin(45) = 69.296 N
Now, let's calculate the net force acting on the mass:
F_net = Component of weight along the incline (mg sinθ) - Friction force (F_friction)
F_net = 69.296 N - 3.2 N = 66.096 N
Finally, we can use the formula F_net = m * a to calculate the acceleration:
66.096 N = 10 kg * a
a = 6.61 m/s^2
Therefore, the acceleration of the mass is 6.61 m/s^2.
To find the acceleration of the mass sliding down the inclined plane, we can use Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration.
First, we need to break down the forces acting on the object along the inclined plane.
1. The force due to gravity: The weight of the object is given by the formula W = m * g, where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s^2). In this case, the mass is 10 kg, so the force due to gravity is W = 10 kg * 9.8 m/s^2 = 98 N.
2. The force opposing motion: The frictional force opposing the motion of the object is given as 3.2 N. Since the object is sliding down the plane, the frictional force acts in the opposite direction to the motion.
Next, we need to resolve the forces acting on the object along the vertical and horizontal directions:
1. Vertical direction: The force due to gravity can be resolved into two components: the force acting perpendicular to the plane (mg * cos θ) and the force acting parallel to the plane (mg * sin θ). Here, θ is the inclination angle of the plane, which is 45 degrees. So, the force acting perpendicular to the plane is:
F_perpendicular = mg * cos θ = 10 kg * 9.8 m/s^2 * cos(45°) ≈ 68.68 N.
2. Horizontal direction: The force acting parallel to the plane is responsible for the acceleration of the object. Considering the positive direction as the direction of motion, the force opposing motion is in the negative direction. Hence, the net force acting on the object in the horizontal direction is:
F_net = mg * sin θ - frictional force = 10 kg * 9.8 m/s^2 * sin(45°) - 3.2 N.
Using Newton's second law, we equate the net force to the mass of the object multiplied by its acceleration:
F_net = m * a,
10 kg * 9.8 m/s^2 * sin(45°) - 3.2 N = 10 kg * a.
Simplifying the equation, we can find the value of acceleration (a):
a = (10 kg * 9.8 m/s^2 * sin(45°) - 3.2 N) / 10 kg.
Calculating this expression, we get:
a ≈ (98 * √(2) - 3.2) / 10 ≈ 13.83 m/s^2.
Therefore, the acceleration of the mass sliding down the inclined plane, given a frictional force of 3.2 N, is approximately 13.83 m/s^2.