A small block of mass is released from rest at the top of a frictionless incline that…
PHYSICS · Multi-part · CBSE · Physics · Laws of Motion · Laws of Motion
Problem
A small block of mass m is released from rest at the top of a frictionless incline that makes an angle θ with the horizontal. Two classmates, Alex and Brianna, each sketch a free-body diagram (FBD) for the block while it is sliding down the plane.
Alex’s diagram (Diagram A):
• Central dot.
• Arrow labeled mg drawn straight downward.
• Arrow labeled N perpendicular to and away from the surface of the incline.
• Two additional arrows originating at the dot: one labeled mgcosθ perpendicular to the incline and pointing into the surface, and another labeled mgsinθ pointing down the incline, parallel to the surface.
Brianna’s diagram (Diagram B):
• Central dot.
• Arrow labeled mg drawn straight downward.
• Arrow labeled N perpendicular to and away from the surface of the incline.
• No other arrows.
Assume the block is treated as a single object and air resistance is negligible.
(a) Indicate which diagram correctly represents the block’s free-body diagram. Justify your answer by referencing the defining features of a proper FBD.
(b) For the diagram you judged to be incorrect, list every specific error it contains and explain how each error violates FBD conventions or physical principles.
(c) Using only the forces that appear on the correct FBD, derive an expression for the magnitude of the block’s acceleration along the incline in terms of g and θ. Then calculate the acceleration for θ=30∘. Show all work.