What is the tension in a rope that is being used to pull a crate along a frictionlesshorizontal surface at a rate of 4 m/s2? The mass of the crate is 1200 kg. The anglebetween the rope and the horizontal is 30ºA) 12000 NB) 5543 NC) 14200 ND) 6200 NE) 4800 N
Question
What is the tension in a rope that is being used to pull a crate along a frictionlesshorizontal surface at a rate of 4 m/s2? The mass of the crate is 1200 kg. The anglebetween the rope and the horizontal is 30ºA) 12000 NB) 5543 NC) 14200 ND) 6200 NE) 4800 N
Solution 1
Para resolver este problema, debemos descomponer las fuerzas involucradas y aplicar la segunda ley de Newton. Aquí están los pasos detallados:
-
Identificar las fuerzas:
- La fuerza de tensión en la cuerda (T).
- La componente horizontal de la tensión (T_x).
- La componente vertical de la tensión (T_y).
- La fuerza de gravedad (mg).
-
Descomponer la tensión en componentes:
- T_x = T * cos(30º)
- T_y = T * sin(30º)
-
Aplicar la segunda ley de Newton en la dirección horizontal:
- La aceleración horizontal es 4 m/s².
- La masa del cajón es 1200 kg.
- La fuerza neta en la dirección horizontal es F_net = m * a = 1200 kg * 4 m/s² = 4800 N.
-
Relacionar la fuerza neta con la componente horizontal de la tensión:
- T_x = F_net
- T * cos(30º) = 4800 N
-
Resolver para la tensión (T):
- T = 4800 N / cos(30º)
- cos(30º) = √3 / 2 ≈ 0.866
- T = 4800 N / 0.866 ≈ 5543 N
Por lo tanto, la tensión en la cuerda es aproximadamente 5543 N.
La respuesta correcta es: B) 5543 N
Solution 2
Para resolver este problema, debemos descomponer las fuerzas involucradas y aplicar la segunda ley de Newton. Aquí están los pasos detallados:
-
Identificar las fuerzas:
- La fuerza de tensión en la cuerda (T).
- La componente horizontal de la tensión (T_x).
- La componente vertical de la tensión (T_y).
- La fuerza de gravedad (mg).
-
Descomponer la tensión en componentes:
- T_x = T * cos(30º)
- T_y = T * sin(30º)
-
Aplicar la segunda ley de Newton en la dirección horizontal:
- La aceleración horizontal es 4 m/s².
- La masa del cajón es 1200 kg.
- La fuerza neta en la dirección horizontal es F_net = m * a = 1200 kg * 4 m/s² = 4800 N.
-
Relacionar la fuerza neta con la componente horizontal de la tensión:
- T_x = F_net
- T * cos(30º) = 4800 N
-
Resolver para la tensión (T):
- T = 4800 N / cos(30º)
- cos(30º) = √3 / 2 ≈ 0.866
- T = 4800 N / 0.866 ≈ 5543 N
Por lo tanto, la tensión en la cuerda es aproximadamente 5543 N.
La respuesta correcta es: B) 5543 N
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