A light rope is attached to a block with mass $4.00$ kg that rests on a frictionless, horizontal surface. The horizontal rope passes over a frictionless, massless pulley, and a block with mass $m$ is suspended from the other end. When the blocks are released, the tension in the rope is $15.0$ N. What is the acceleration of either block?

검증된 단계별 안내

Identify the forces acting on each block. For the 4.00 kg block on the horizontal surface, the forces are the tension in the rope (T = 15.0 N) and the normal force balancing its weight. For the hanging block with mass m, the forces are the tension in the rope (T = 15.0 N) acting upward and the gravitational force (m * g) acting downward.

Write Newton's second law for the 4.00 kg block on the horizontal surface. Since it is accelerating to the right, the net force is equal to the tension in the rope: F_net = T = m₁ * a, where m₁ = 4.00 kg and a is the acceleration.

Write Newton's second law for the hanging block. The net force is the difference between the gravitational force and the tension: F_net = m * g - T = m₂ * a, where m₂ is the mass of the hanging block, g is the acceleration due to gravity (9.8 m/s²), and a is the acceleration.

Combine the two equations from steps 2 and 3. Solve for the acceleration (a) by eliminating the tension (T). From step 2: a = T / m₁. Substitute this into the equation from step 3: m * g - T = m₂ * (T / m₁). Rearrange to isolate a.

Simplify the resulting equation to find the acceleration (a) in terms of the given values (T = 15.0 N, m₁ = 4.00 kg, g = 9.8 m/s²). This will give you the acceleration of both blocks, as they are connected by the rope and must have the same magnitude of acceleration.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed by the formula F = ma, where F is the net force, m is the mass, and a is the acceleration. In this scenario, the tension in the rope and the weight of the suspended block create the net force that determines the acceleration of both blocks.

Tension in a Rope

Tension is the force transmitted through a rope or string when it is pulled tight by forces acting at either end. In this problem, the tension in the rope affects both the block on the frictionless surface and the suspended block. Understanding how tension works is crucial for analyzing the forces acting on each block and calculating their acceleration.

Free Body Diagram

A Free Body Diagram (FBD) is a graphical representation used to visualize the forces acting on an object. By isolating the blocks and drawing FBDs, one can identify all the forces, including tension and gravitational force, acting on each block. This helps in applying Newton's laws to solve for unknown quantities, such as acceleration in this case.