6.4
Segunda Lei: Movimento sob a Mesma Aceleração
A segunda lei do movimento de Newton se aplica a corpos em movimento sob a mesma aceleração. Por exemplo, quando um trator de bagagens puxa carrinhos…
Corpos que se movem com a mesma aceleração obedecem à segunda lei de Newton. Considere um carro rebocando outro carro idêntico com uma corda inextensível sem massa. A corda aplica força igual e oposta em ambos os carros. Então, por que o carro deveria acelerar?
Para responder a isso, um diagrama de corpo livre de cada um é desenhado. Como as forças verticais são equilibradas, a força resultante que atua em cada uma é obtida usando os componentes de força horizontal.
Agora, usando o fato de que ambos os carros têm a mesma massa e aceleração, as equações são equiparadas para obter a força que faz com que os carros acelerem.
Considere outro exemplo de uma corda inextensível e sem massa pendurada em duas massas desiguais sobre uma polia sem atrito. Quando o sistema é liberado, a massa mais pesada levanta a mais leve, e ambos se movem com a mesma aceleração.
Usando diagramas de corpo livre e a segunda lei de Newton, uma equação para a força resultante em cada massa pode ser derivada. Ao subtrair essas equações e reorganizá-las, obtém-se o valor da aceleração das massas.
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Q1: Why do two cars connected by a rope accelerate together?
When a car tows another identical car with a massless inextensible rope, both experience the same acceleration because the rope applies equal and opposite forces on each. Using free-body diagrams, the net horizontal force on each car equals its mass times acceleration. Since both cars have identical mass and acceleration, they move together as a system.
Q2: How do you find acceleration when unequal masses hang over a pulley?
Draw free-body diagrams for each mass and apply Newton's second law separately. The heavier mass pulls the lighter one upward while descending. Both masses move with the same acceleration despite their different weights. Subtract the force equations and rearrange to solve for the acceleration value.
Q3: What role does tension play in a string connecting two blocks?
Tension is the force transmitted through a massless string connecting two blocks. The tension remains constant throughout the string and acts horizontally on a block on a surface and vertically on a hanging block. Tension pulls each block in opposite directions, enabling them to move with the same acceleration.
Q4: Why do blocks on a horizontal surface and hanging blocks have equal acceleration?
When one block moves horizontally on a frictionless surface and another hangs vertically, connected by a string over a pulley, they travel equal distances in equal times. This constraint means both blocks must have the same acceleration magnitude. Newton's second law applied to each block confirms this kinematic relationship.
Q5: How do free-body diagrams help solve problems with same acceleration?
Free-body diagrams isolate each object and show all forces acting on it. For objects with the same acceleration, you write component equations for horizontal and vertical forces separately. Balanced forces in one direction simplify the analysis, allowing you to focus on net forces causing acceleration.
Q6: What happens to tension when two blocks of different masses hang vertically?
When unequal masses hang vertically over a frictionless pulley, the tension in the string is less than the weight of the heavier mass but greater than the weight of the lighter mass. The tension accelerates the lighter mass upward and reduces the net downward force on the heavier mass, allowing both to accelerate together.
Q7: Why are vertical and horizontal forces analyzed separately in these problems?
Forces perpendicular to each other do not affect one another. On a horizontal surface, vertical forces balance, so only horizontal components determine acceleration. For hanging blocks, only vertical forces matter. Separating components simplifies calculations and reveals which forces actually cause the observed acceleration.