A block with mass M slides down an incline of angle θ at constant velocity v. Which of the following expressions is equal to the magnitude of frictional force between the block and the incline? (Note: The acceleration of gravity g = 10 m/s².)

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11ecba2d_131a_a2d5_a3bf_c562407588ac_MD0008_00 Multiple translational forces, each with unique magnitude and direction, may simultaneously act on the same object. According to Newton's second law, an object will accelerate if the net force acting along each axis of the x,y-coordinate system is a nonzero quantity. Conversely, equilibrium occurs when the net force acting along each axis of the x,y-coordinate system equals zero such that the object does not accelerate. Dynamic equilibrium is a special type of equilibrium that describes an object moving at a constant velocity without acceleration. Like other forms of equilibria, dynamic equilibrium may result from either the absence of translational forces or a perfect balance between antagonistic forces. Provided that the x- and y-components of each force vector are known, free-body diagrams can be used to assess equilibrium resulting from multiple forces, where: 11ecba2d_131a_c9e6_a3bf_99a81dd76ad7_MD0008_00 11ecba2d_131a_c9e7_a3bf_f32674b3f74d_MD0008_00 For example, a mass sliding down an incline at constant velocity is acted on by multiple forces. To simplify the analysis of the forces, the vector components of force can be classified as acting either parallel (||) or perpendicular (⊥) to the surface of the incline, such that: 11ecba2d_131a_c9e8_a3bf_4139627e2174_MD0008_00 11ecba2d_131a_c9e9_a3bf_e954949ef349_MD0008_00 Accordingly, for a block to slide down an incline with nonzero friction at constant velocity requires that the frictional force (F_f) resisting motion be equal and opposite to the component of the gravitational force that acts parallel to the surface of the incline (F_||): 11ecba2d_131a_c9ea_a3bf_5932ccf2ebe5_MD0008_00 11ecba2d_131a_c9eb_a3bf_5b382839bee2_MD0008_00 Furthermore, the component of gravitational force that acts parallel to the surface of the incline may be expressed in terms of the weight of the object (M∙g) and the angle of incline (θ) as: 11ecba2d_131a_f0fc_a3bf_e14026c3cd39_MD0008_00 11ecba2d_131a_f0fd_a3bf_bd93ccef06b4_MD0008_00 Therefore, the frictional force acting on a mass sliding down an incline in dynamic equilibrium is equal in magnitude to the weight of the object multiplied by the sine of the incline angle (Choice A). Educational objective: Dynamic equilibrium occurs when the sum of all forces acting on an object in motion is equal to zero (ie, the object moves but does not accelerate). Free-body diagrams are used to assess the directional components of translational forces.

A Block with Mass M Slides Down an Incline of Angle