A 50kg box is placed on an incline that makes an angle of 30° with respect to the horizontal.The coefficient of friction between the mass and the incline is 0.2. Find the acceleration of thebox going down the incline. (2 sig-figs)
Question
A 50kg box is placed on an incline that makes an angle of 30° with respect to the horizontal.The coefficient of friction between the mass and the incline is 0.2. Find the acceleration of thebox going down the incline. (2 sig-figs)
Solution
Sure, let's solve this step by step:
Step 1: Identify the forces acting on the box There are three forces acting on the box:
- The force of gravity (Fg), which acts vertically downward
- The normal force (Fn), which acts perpendicular to the incline
- The frictional force (Ff), which acts parallel to the incline and opposes the motion of the box
Step 2: Calculate the force of gravity (Fg) The force of gravity is given by the equation Fg = m*g, where m is the mass of the box and g is the acceleration due to gravity. So, Fg = 50 kg * 9.8 m/s^2 = 490 N
Step 3: Calculate the component of the gravitational force parallel to the incline (Fg_parallel) This is given by the equation Fg_parallel = Fg * sin(θ), where θ is the angle of the incline. So, Fg_parallel = 490 N * sin(30°) = 245 N
Step 4: Calculate the component of the gravitational force perpendicular to the incline (Fg_perpendicular) This is given by the equation Fg_perpendicular = Fg * cos(θ). So, Fg_perpendicular = 490 N * cos(30°) = 424.26 N
Step 5: Calculate the frictional force (Ff) The frictional force is given by the equation Ff = μ * Fn, where μ is the coefficient of friction and Fn is the normal force. In this case, the normal force is equal to the component of the gravitational force perpendicular to the incline (Fg_perpendicular). So, Ff = 0.2 * 424.26 N = 84.85 N
Step 6: Calculate the net force acting on the box (Fnet) The net force is the difference between the component of the gravitational force parallel to the incline and the frictional force. So, Fnet = Fg_parallel - Ff = 245 N - 84.85 N = 160.15 N
Step 7: Calculate the acceleration of the box (a) The acceleration is given by the equation a = Fnet / m. So, a = 160.15 N / 50 kg = 3.20 m/s^2
Therefore, the acceleration of the box going down the incline is approximately 3.20 m/s^2.
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