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Consider the incline shown in the figure with inclination angle θ and height h. The coefficient of kinetic friction between a block of mass m and the incline changes along its surface. From A to B it has a value of μ1, and from B to C its value is μ2. The block (not shown) is given an initial velocity up the incline at point A with a magnitude vA.   A) What is the value of vA for which the block will reach the top of the incline (point C) with zero kinetic energy? Your answer should be written in terms of the quantities given in the statement.   B) Show that for the block to surpass point B, the initial speed at A should satisfy the inequality  vA > √[ (2/3)gh(1+μ1cotθ) ]

Consider the incline shown in the figure with inclination angle θ and height h. The coefficient of kinetic friction between a block of mass m and the incline changes along its surface. From A to B it has a value of μ1, and from B to C its value is μ2. The block (not shown) is given an initial velocity up the incline at point A with a magnitude vA.   A) What is the value of vA for which the block will reach the top of the incline (point C) with zero kinetic energy? Your answer should be written in terms of the quantities given in the statement.   B) Show that for the block to surpass point B, the initial speed at A should satisfy the inequality  vA > √[ (2/3)gh(1+μ1cotθ) ]

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Consider the incline shown in the figure with inclination angle θ and height h. The coefficient of kinetic friction between a block of mass m and the incline changes along its surface. From A to B it has a value of μ1, and from B to C its value is μ2. The block (not shown) is given an initial velocity up the incline at point A with a magnitude vA.

 

A) What is the value of vA for which the block will reach the top of the incline (point C) with zero kinetic energy? Your answer should be written in terms of the quantities given in the statement.

 

B) Show that for the block to surpass point B, the initial speed at A should satisfy the inequality 

vA > √[ (2/3)gh(1+μ1cotθ) ]

This diagram represents a right triangle \( \triangle ABC \). - **Vertices:** - \( A \) is at the left-bottom corner of the triangle. - \( B \) is the point where the hypotenuse meets the horizontal line segment \( AB \). - \( C \) is at the top of the perpendicular, forming the right angle with line segment \( AB \). - **Sides:** - \( AB \) is the horizontal line segment. - \( BC \) is the hypotenuse of the triangle. - \( AC \) is the vertical line segment representing the height. - **Angles:** - \( \theta \) is marked at vertex \( A \), representing the angle between side \( AB \) and the hypotenuse \( BC \). - **Heights and Notations:** - The perpendicular height from \( A \) to \( C \) is labeled as \( h \). - A horizontal dotted line extends from point \( B \) to the vertical line from \( C \) to \( A \), indicating a segment marked as \( h/3 \). - The side \( BC \) is labeled with the symbol \( \mu_c \). - A segment of \( AB \), from \( A \) to \( B \), is labeled as \( \mu_l \). This diagram could be used to illustrate concepts in trigonometry, such as calculating angles, side lengths using trigonometric ratios, or demonstrating the properties of angles and their corresponding sides in a right triangle.
Transcribed Image Text:This diagram represents a right triangle \( \triangle ABC \). - **Vertices:** - \( A \) is at the left-bottom corner of the triangle. - \( B \) is the point where the hypotenuse meets the horizontal line segment \( AB \). - \( C \) is at the top of the perpendicular, forming the right angle with line segment \( AB \). - **Sides:** - \( AB \) is the horizontal line segment. - \( BC \) is the hypotenuse of the triangle. - \( AC \) is the vertical line segment representing the height. - **Angles:** - \( \theta \) is marked at vertex \( A \), representing the angle between side \( AB \) and the hypotenuse \( BC \). - **Heights and Notations:** - The perpendicular height from \( A \) to \( C \) is labeled as \( h \). - A horizontal dotted line extends from point \( B \) to the vertical line from \( C \) to \( A \), indicating a segment marked as \( h/3 \). - The side \( BC \) is labeled with the symbol \( \mu_c \). - A segment of \( AB \), from \( A \) to \( B \), is labeled as \( \mu_l \). This diagram could be used to illustrate concepts in trigonometry, such as calculating angles, side lengths using trigonometric ratios, or demonstrating the properties of angles and their corresponding sides in a right triangle.
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