Explanation Given: The roller coaster starts from rest. The normal force is equal to zero or more than 4 times their weight. The radius of curvature at point B is ρ B = 15 m . The radius of curvature at point C is ρ C = 20 m . Draw the free body diagram of passenger at point B and C as shown in Figure (1). Refer Figure (1). Resolve the forces along n − axis to determine the normal reaction force ( N ) . ∑ F n = m a n (I) Here, m is the mass and a n is the acceleration along axis − n . When the roller coaster travel in a curved path it will experience the centripetal acceleration. The acceleration of the path is a n = v 2 ρ . Substitute v 2 ρ for a n in Equation (I). ∑ F n = m ( v 2 ρ ) (II) Write the formula to calculate the work done by the passenger’s weight ( W ) . U W ( 1 − 2 ) = m g h (III) Here, m is the mass, g is acceleration due to gravity and h is the vertical displacement. Write the formula for kinetic energy ( T ) . T = 1 2 m v 2 (IV) Here, m is the mass and v is the velocity. Write the formula for Principle of Work and Energy. T i n i t i a l + ∑ U = T f i n a l (V) Here, T i n i t i a l is initial kinetic energy, T f i n a l is final kinetic energy and ∑ U is total work done by all the forces. Conclusion: Refer Figure (1a). Calculate the normal force acting on the passenger at point B . At point B , the normal force N B = 4 m g . Resolve the forces along n − axis and substitute in Equation (II). N B − m g = m ( v 2 ρ ) Here, v = v B . Substitute 4 m g for N B , v B for v , 15 m for ρ . 4 m g − m g = m ( v B 2 15 m ) m g ( 4 − 1 ) = m v B 2 15 3 g = v B 2 15 v B 2 = 45 g Refer Figure (1b). Calculate the normal force acting on the passenger at point C . At point C , the normal force N C = 0 . Resolve the forces along n − axis and substitute in Equation (II). m g − N C = m ( v 2 ρ ) Here, v = v C . Substitute 0 for N C , v B for v , 20 m for ρ . m g − 0 = m ( v C 2 20 m ) m g = m v C 2 20 g = v C 2 20 v C 2 = 20 g When the roller coaster travels from point A to B

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 14.3, Problem 37P

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Chapter 14.3, Problem 36P

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Chapter 14 Solutions

Engineering Mechanics: Statics & Dynamics (14th Edition)

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The limiting heights h A and h C .

Solution Summary: The author explains how to determine the limiting heights of the roller coaster and the normal force along n-axis.

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The limiting heights hA and hC .

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Chapter 14 Solutions