Question

A mass, \(m_{1}=20.0 \mathrm{~kg},\) on a frictionless ramp is attached to a light string. The string passes over a frictionless pulley and is attached to a hanging mass, \(m_{2}\) The ramp is at an angle of \(\theta=30.0^{\circ}\) above the horizontal. The mass \(m_{1}\), moves up the ramp uniformly (at constant speed). Find the value of \(m_{2}\)

Short answer

Short Answer: To maintain a constant speed of mass \(m_1\) moving up the frictionless ramp, the hanging mass \(m_2\) must be \(10.0 \mathrm{~kg}\). This allows the tension forces acting on the system to balance, and the net force to be zero, satisfying Newton's second law.

Step-by-step solution

Analyze the Forces acting on \(m_{1}\) and \(m_{2}\)

We have two main forces acting on the system: gravitational force and tension. The gravitational force on mass \(m_1\) can be broken into two components: \(m_1g\sin\theta\) along the ramp, and \(m_1g\cos\theta\) perpendicular to the ramp. The force acting on mass \(m_2\) is the gravitational force \(m_2g\). The tension in the string opposes the gravitational force on both \(m_1\) and \(m_2\).

Apply Newton's Second Law to Mass \(m_{1}\)

As the mass \(m_{1}\) is moving at constant speed, the acceleration along the ramp is zero. Applying Newton's second law along the ramp: \(T - m_1g\sin\theta = m_1a\) \(T = m_1g\sin\theta\)

Apply Newton's Second Law to Mass \(m_{2}\)

As the mass \(m_{1}\) is moving at constant speed, the acceleration of mass \(m_{2}\) must also be zero. Applying Newton's second law to mass \(m_{2}\): \(T - m_2g = m_2a\) \(T = m_2g\)

Equate Tensions and Solve for \(m_2\)

From step 2 and step 3, the tension acting on both masses is equal: \(m_1g\sin\theta = m_2g\) Now we can solve for \(m_2\): \(m_2 = \frac{m_1g\sin\theta}{g}\) Plugging in the given values: \(m_2 = \frac{20.0 \mathrm{~kg} \times 9.81 \mathrm{~m/s^2} \times \sin{30^{\circ}}}{9.81 \mathrm{~m/s^2}}\)

Calculate the Value of \(m_{2}\)

Calculate the value of \(m_2\): \(m_2 = \frac{20.0 \mathrm{~kg} \times 9.81 \mathrm{~m/s^2} \times 0.5}{9.81 \mathrm{~m/s^2}}\) \(m_2 = 10.0 \mathrm{~kg}\) Thus, the value of \(m_{2}\) is \(10.0 \mathrm{~kg}\).

Key concepts

Gravitational Force

The force of gravity is a fundamental interaction in the universe affecting all objects with mass. Its significance in physics cannot be overstated. When dealing with problems such as a mass on a ramp, it's essential to understand how gravitational force, represented by the symbol g (with a value of approximately 9.81 m/s² on Earth), influences the movement of objects.

Gravitational force acts directly downwards, towards the center of the Earth. However, when an object is on an inclined plane, like our mass m₁, this force can be decomposed into two components. One component is parallel to the incline (down the ramp), calculated as m₁g sin(θ), and the other is perpendicular to the incline. The parallel component is what drives the object down the slope if unopposed. Since our ramp is frictionless, the only opposing force is the tension in the string connected to mass m₂.

Tension in Physics

In understanding physical systems, tension is as crucial as gravitational force. It's the force that's transmitted through a string, cable, or any other similar object when it's pulled tight by forces acting from opposite ends. In our problem, tension is what connects the two masses across the pulley.

In a real-world scenario, tension would vary along the string due to factors like weight and friction. However, in our idealized exercise, the string is considered light (massless) and the pulley frictionless, meaning the tension is constant throughout the string. This simplification is key, as it allows us to equate the force of tension T holding up mass m₂ with the force pulling mass m₁ up the incline. This understanding helps solve for unknowns in the system using the equations provided in the solution steps.

Frictionless Ramp Problem

A frictionless ramp problem represents an ideal physics model where an object slides down an inclined surface without the resistance caused by friction. These problems are ideal starting points for learning physics concepts since they focus on the basics of motion and forces without the added complexity that friction brings.

For an object moving on a frictionless ramp, as in our exercise, only gravity and tension are at play. Since there's no friction to oppose the motion, if the object is moving at a constant speed (no acceleration), then the forces parallel to the ramp's surface must be balanced. This means the component of gravitational force acting down the ramp is equal in magnitude to the tension force acting up the ramp. Understanding this balance of forces is crucial to solving ramp problems and is a great example of Newton's Second Law in action, stating that F = ma. In this case, a is zero, so the forces must be in equilibrium.