Question

A hanging mass, \(M_{1}=0.400 \mathrm{~kg}\), is attached by a light string that runs over a frictionless pulley to a mass \(M_{2}=1.20 \mathrm{~kg}\) that is initially at rest on a frictionless ramp. The ramp is at an angle of \(\theta=30.0^{\circ}\) above the horizontal, and the pulley is at the top of the ramp. Find the magnitude and direction of the acceleration, \(a_{2},\) of \(M_{2}\)

Short answer

Question: Calculate the acceleration of mass M2 and determine its direction. Answer: The acceleration of mass M2 is 3.27 m/s², and its direction is down the inclined plane.

Step-by-step solution

Identifying the forces acting on the masses

We'll first identify the forces acting on both masses and make a free body diagram. For M1, there is tension, T, in the string and gravitational force (weight), \(M_{1}g\). For M2, there is gravitational force (weight), \(M_{2}g\), tension, T, and the normal force, N. Note that the gravitational force on M2 can be decomposed into two components, i.e. \(M_{2}g\sin\theta\) and \(M_{2}g\cos\theta\) parallel and perpendicular to the inclined plane, respectively.

Setting up the motion equations

Write the equations of motion for M1 and M2 based on Newton's second law, \(F=ma\). For M1 (vertically downward direction): $$T - M_{1}g = M_{1}a_{1}$$ For M2 (along the inclined plane in down-direction being positive): $$M_{2}g\sin\theta - T = M_{2}a_{2}$$ Note that because the string connecting the masses is light and inextensible, \(|a_{1}| = |a_{2}|\). Thus, we can use \(a\) to represent the magnitudes of both accelerations.

Solving for the acceleration

Now, we want to solve for the acceleration \(a\). Eliminate tension T from the equations by adding them together: $$M_{2}g\sin\theta - M_{1}g = (M_{1}+ M_{2})a$$ Rearrange the formula and solve for \(a\): $$a = \frac{M_{2}g\sin\theta - M_{1}g}{M_{1}+ M_{2}}$$ Plug in the given values for the masses and the angle: $$a = \frac{1.20(9.8)\sin 30^{\circ} - 0.400(9.8)}{0.400+ 1.20}$$ Calculate the acceleration: $$a = 3.27\ \mathrm{m/s^2}$$

Determine the direction

Now that we've found the magnitude of the acceleration, let's determine its direction. As \(M_{2}g\sin\theta > T\), the acceleration \(a_{2}\) is directed down the ramp. Therefore, the direction of the acceleration, \(a_{2}\), is down the inclined plane.

Final Answer

The magnitude of the acceleration, \(a_{2}\), is \(3.27\ \mathrm{m/s^2}\), and the direction is down the inclined plane.

Key concepts

Newton's Second Law

Newton's second law of motion is the cornerstone when it comes to understanding motion in classical mechanics. It defines the relationship between the net force acting on an object and its acceleration. The law is often stated as \( F = ma \), where \( F \) is the net force acting on the object, \( m \) is the mass of the object, and \( a \) is the acceleration.

In our inclined plane problem, Newton's second law helps us determine the acceleration of the masses involved. Because the only forces acting on the masses are gravity, tension, and the normal force, we can apply the law to find out how these forces affect the system's movement.

By breaking down the forces individually for each object—mass \( M_1 \) being affected by gravity and tension, and mass \( M_2 \) by gravity, tension, and the normal force—we are able to establish equations that will ultimately allow us to solve for the acceleration of the system.

Free Body Diagrams

A free body diagram is a powerful tool in physics that allows us to visually represent the forces acting on an object. This diagram will enable us to isolate a single body and depict all external forces acting upon it, thereby simplifying the problem-solving process.

In our inclined plane example, we can create separate free body diagrams for each mass. For mass \( M_1 \), the free body diagram would show the downward gravitational force and the upwards tension force in the string. Conversely, for mass \( M_2 \) on the inclined plane, we would show three forces: the gravitational force pulling it down the slope (\( M_{2}g\sin θ \)), the component of the gravitational force acting perpendicular to the slope (\( M_{2}g\cos θ \)), tension up the slope, and the normal force acting perpendicular to the surface upwards.

These diagrams are essential as they lead us to set up the correct equations for Newton's second law, guiding our understanding of the physical situation and assisting in accurately assessing forces.

Acceleration Calculation

Now, we come to the actual computation of acceleration in an inclined plane problem. After applying Newton's second law to set up our motion equations, we're able to solve for acceleration by manipulating these equations.

As shown in the solution steps, by adding the motion equations we have for \( M_1 \) and \( M_2 \) and eliminating the tension, we obtain a formula that allows us to directly compute the acceleration of the system. The formula we derive encapsulates how the interplay between the gravitational force and the physical configuration of the system (like the incline angle) determines acceleration.

\[ a = \frac{M_{2}g\sin θ - M_{1}g}{M_{1} + M_{2}} \]
After substituting all known values into the derived formula, we get a numerical value for acceleration. This assists us in understanding not just how fast the object will move, but also how movement dynamics on an inclined plane are distinctly different from those on a flat surface.