Questions: A piano has been pushed to the top of the ramp at the back of a moving van. The workers think it is safe, but as they walk away, it begins to roll down the ramp. Neglect the friction between the piano and the ramp.

A piano has been pushed to the top of the ramp at the back of a moving van. The workers think it is safe, but as they walk away, it begins to roll down the ramp. Neglect the friction between the piano and the ramp.

Transcript text: A piano has been pushed to the top of the ramp at the back of a moving van. The workers think it is safe, but as they walk away, it begins to roll down the ramp. Neglect the friction between the piano and the ramp.

Solution

Solution Steps

Step 1: Understanding the Problem

The problem involves a piano rolling down a ramp due to gravity. We are asked to analyze the motion of the piano as it rolls down the ramp. The key point is that friction is neglected, meaning the only force acting on the piano is gravity.

Step 2: Analyzing Forces

Since friction is neglected, the only force acting on the piano is the component of gravitational force along the ramp. This force can be calculated using the formula: \[ F_{\text{gravity, parallel}} = m \cdot g \cdot \sin(\theta) \] where \( m \) is the mass of the piano, \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)), and \( \theta \) is the angle of the ramp with respect to the horizontal.

Step 3: Calculating Acceleration

The acceleration of the piano down the ramp can be found using Newton's second law: \[ F = m \cdot a \] Substituting the force from Step 2, we get: \[ m \cdot a = m \cdot g \cdot \sin(\theta) \] Solving for acceleration \( a \), we find: \[ a = g \cdot \sin(\theta) \]