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. (b) What is the acceleration of either block?

Verified step by step guidance

1. Identify the forces acting on each block. For the 4.00 kg block, the only horizontal force is the tension T. For the block with mass m, the forces are the tension T upwards and the gravitational force mg downwards.
2. Apply Newton's second law to each block. For the 4.00 kg block, the equation is T = 4.00a, where a is the acceleration. For the block with mass m, the equation is T - mg = ma.
3. Substitute the given tension T = 15.0 N into the equations. This gives 15.0 = 4.00a for the 4.00 kg block and 15.0 - mg = ma for the block with mass m.
4. Solve the first equation for a to find the acceleration of the 4.00 kg block: a = 15.0 / 4.00.
5. Use the value of a from step 4 in the second equation to find the mass m of the other block. Substitute a into the equation 15.0 - mg = ma and solve for m.

Key Concepts

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 surface and the suspended block. Since the pulley is frictionless and massless, the tension remains constant throughout the rope, allowing us to analyze the forces acting on both blocks to find their acceleration.

Free Body Diagram

A Free Body Diagram (FBD) is a graphical representation used to visualize the forces acting on an object. In this case, drawing FBDs for both blocks helps identify the forces at play, such as tension and gravitational force. By applying Newton's Second Law to these diagrams, we can derive equations that relate the masses, tension, and acceleration, ultimately leading to the solution of the problem.