Question

A hanging mass, \(M_{1}=0.500 \mathrm{~kg}\), is attached by a light string that runs over a frictionless pulley to a mass \(M_{2}=1.50 \mathrm{~kg}\) that is initially at rest on a frictionless table. Find the magnitude of the acceleration, \(a\), of \(M_{2}\).

Short answer

Answer: The acceleration of mass M2 is 2.45 m/s².

Step-by-step solution

Set up the problem

Draw the free body diagram of the system, which includes the two masses, the forces acting on them (weight and tension), and the pulley. Specify the coordinate systems for each mass.

Write Newton's second law of motion for each mass

For mass \(M_1\), we have: $$M_1g - T = M_1a$$ For mass \(M_2\), we have: $$T = M_2a$$

Solve for the tension T

Solving the equation for mass \(M_2\) for tension \(T\), we get: $$T = M_2a$$

Substitute the T expression into the equation for mass M1

Now we can substitute the expression for the tension \(T\) from Step 3 into the equation for mass \(M_1\): $$M_1g - M_2a = M_1a$$

Solve for acceleration a

Rearrange the last equation to solve for acceleration \(a\): $$M_1g = (M_1+M_2) a$$ Divide both sides by (\(M_1\)+\(M_2\)) to get: $$a = \frac{M_1g}{(M_1+M_2)}$$

Calculate the acceleration using the given values

Now we can substitute the given values of the masses into the expression for acceleration: \(a = \frac{M_1g}{(M_1+M_2)} = \frac{(0.500 \mathrm{~kg})(9.81 \mathrm{~m/s^2})}{(0.500 \mathrm{~kg} + 1.50 \mathrm{~kg})}\) \(a = \frac{4.905 \mathrm{~kg\cdot m/s^2}}{2.00 \mathrm{~kg}}\)

Calculate the final value of acceleration

Finally, divide the numerator by the denominator to find the acceleration: \(a = 2.45 \mathrm{~m/s^2}\) The acceleration of mass \(M_2\) is 2.45 m/s².

Key concepts

Free Body Diagram

A free body diagram is a simple yet powerful tool used to show all the forces acting upon a single object or system of objects. Imagine you have a complex problem involving multiple forces; the free body diagram simplifies this problem by representing each force with an arrow pointing in the direction the force is applied. The length of the arrow generally depicts the magnitude of the force. In the case of our hanging mass system, we would draw separate diagrams for each mass. For mass A (hanging mass), we have gravity acting downwards and the tension in the string acting upwards. For mass B (mass on the table), we only have the tension acting horizontally since the mass is not accelerating vertically, and there is no friction due to the frictionless surface.

To enhance understanding and avoid mistakes, ensure that your free body diagrams are drawn neatly, with forces labeled correctly, and the object isolated from its environment. This visualization technique is a cornerstone of solving problems in classical mechanics because it clarifies forces and accelerations, making subsequent calculations more straightforward.

Frictionless Pulley System

In physics problems, a pulley system usually introduces complexity, but the assumption of a frictionless pulley greatly simplifies the analysis. A frictionless pulley means that the pulley does not resist the movement of the string, so there's no energy loss due to heat, and the tension remains the same throughout the string. This means that the pulley changes the direction of the tension force without changing its magnitude. In our hanging mass problem, the frictionless assumption allows us to equate the tension on both sides of the pulley. Without friction, the only forces we need to consider for the mass on the table are the tension and the normal force (which is balanced by the weight and thus not a factor in horizontal motion).

When drawing the system or free body diagrams, remember to indicate that the pulley is frictionless to signify that it does not affect the tension in the string or contribute any additional forces.

Solving for Acceleration

Solving for acceleration is a crucial part of understanding dynamic systems in physics. Acceleration tells us how quickly the velocity of an object is changing and is a direct consequence of Newton's second law, which states that force equals mass times acceleration (\( F = ma \)). To find the acceleration in our problem, we applied Newton’s second law to each mass separately and then combined these equations to eliminate the unknown tension.

To tackle similar problems, always start by identifying all forces and write down Newton's second law for each object. Then, use algebra to solve the system of equations. Remember that understanding how to manipulate these equations is just as essential as understanding the physical concepts behind them. One way to check if your solution is reasonable is to consider the limits of the system — for example, if one mass were infinitely heavy, it would not accelerate, and the acceleration of our system should approach zero.

Tension in Physics

In our textbook problem, tension represents the force conducted along the string that connects the two masses. Tension is a force that is transmitted through a string, rod, cable or other similar objects when they are pulled tight by forces acting from opposite ends. It's always a pulling force and acts along the length of the object transmitting it. In the context of a frictionless pulley system, the tension is the same on both sides of the pulley as there are no other forces along the string to change it, and the pulley does not add or subtract from it due to the lack of friction.

In problems involving multiple objects and strings, remember that the magnitude of tension might change from one segment of the string to another if there are other forces at play, such as friction or if the pulleys are not frictionless. In the case of our mass system, because we're dealing with a frictionless surface and a frictionless pulley, the tension throughout the string remains constant. If additional forces were present, we would need to consider them in our free body diagrams and subsequent calculations.