A 20 kg box is released from rest at point A. It travels down a slope of 35 degrees. If the coefficient of
friction is 30%, what will be the velocity of the box 10 meters from point A.
To find the velocity of the box 10 meters from point A, we need to use the principles of physics.
First, let's break down the problem into several steps:
Step 1: Determine the net force acting on the box.
Step 2: Apply Newton's Second Law to find the acceleration.
Step 3: Use the acceleration to find the final velocity.
Now, let's go through each step:
Step 1: Determine the net force acting on the box.
The net force is the vector sum of all the forces acting on the box. In this case, we have two forces:
1. The force due to gravity, which acts vertically downwards and has a magnitude of Fg = m * g, where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. The force of friction, which opposes the motion of the box down the slope. The frictional force can be calculated as Ff = u * N, where u is the coefficient of friction and N is the normal force. The normal force is the force exerted by the surface on the box and can be found as N = m * g * cosθ, where θ is the angle of the slope.
Step 2: Apply Newton's Second Law to find the acceleration.
Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. So, we have Fnet = m * a, where Fnet is the net force and a is the acceleration. Rearranging the equation, we get a = Fnet / m.
Step 3: Use the acceleration to find the final velocity.
The final velocity can be calculated using the equation v^2 = u^2 + 2 * a * s, where v is the final velocity, u is the initial velocity (which is zero in this case since the box is released from rest), a is the acceleration, and s is the displacement (which is 10 meters in this case).
Now, let's calculate the values step by step:
Step 1: Determine the net force acting on the box.
The force due to gravity is Fg = m * g = 20 kg * 9.8 m/s^2 = 196 N.
The normal force is N = m * g * cos(35 degrees) = 20 kg * 9.8 m/s^2 * cos(35 degrees) ≈ 160.814 N.
The frictional force is Ff = u * N = 0.3 * 160.814 N = 48.2442 N.
The net force is given by Fnet = Fg - Ff = 196 N - 48.2442 N = 147.7558 N.
Step 2: Apply Newton's Second Law to find the acceleration.
a = Fnet / m = 147.7558 N / 20 kg ≈ 7.3878 m/s^2.
Step 3: Use the acceleration to find the final velocity.
v^2 = u^2 + 2 * a * s = 0 + 2 * 7.3878 m/s^2 * 10 m = 147.756 m^2/s^2.
Taking the square root of both sides, we get v ≈ 12.1597 m/s.
Therefore, the velocity of the box 10 meters from point A is approximately 12.16 m/s.