Question

Two blocks, with masses \(m_{1}=4.6 \mathrm{~kg}\) and \(m_{2}=3.8 \mathrm{~kg}\), are connected by a light spring on a horizontal frictionless table. At a certain instant, when \(m_{2}\) has an acceleration \(a_{2}=\) \(2.6 \mathrm{~m} / \mathrm{s}^{2},(a)\) what is the force on \(m_{2}\) and \((b)\) what is the acceleration of \(m_{1}\) ?

Short answer

The force on \(m_{2}\) is \(F = m_{2} \cdot a_{2}\) and the acceleration of \(m_{1}\) is \(a_{1} = F / m_{1}\). Substitute the given values to get the numerical answers.

Step-by-step solution

Calculate the force on \(m_{2}\)

Use Newton's second law of motion, which is \(F=m \cdot a\). The force (\(F\)) acting on second block can be calculated by multiplying mass (\(m_{2}\)) of the block with its acceleration (\(a_{2}\)). So \(F = m_{2} \cdot a_{2} = 3.8 \mathrm{~kg} \cdot 2.6 \mathrm{~m/s^{2}}\).

Calculate the acceleration of \(m_{1}\)

Again by using Newton's second law, we can determine the acceleration (\(a_{1}\)) of block \(m_{1}\) since we know the force acting on it is the same as the force acting on \(m_{2}\). So, \(a_{1} = F / m_{1}\). Replace \(F\) with the value we found in Step 1, and \(m_{1}\) with its given value to obtain \(a_{1}\).

Key concepts

Force Calculation

Understanding how to calculate force is crucial in physics, particularly when applying Newton's second law of motion. This law states that the force acting upon an object is equal to the mass of the object times its acceleration (\( F = m \times a \)). In other words, force is the product of an object's mass and its change in velocity over time.

In the given problem, the force on block with mass \( m_{2} = 3.8 \; \text{kg} \) is found by multiplying its mass by its acceleration \( a_{2} = 2.6 \; \text{m/s}^{2} \). It's essential to ensure that the mass is measured in kilograms (kg) and acceleration in meters per second squared (\(\text{m/s}^2\)) to maintain SI unit consistency. This means the force on \( m_{2} \) can be calculated as \( F = m_{2} \times a_{2} = 3.8 \; \text{kg} \times 2.6 \; \text{m/s}^{2} = 9.88 \; \text{N} \) (where N stands for Newtons). It's the simplicity of this formula that makes force calculation a fundamental skill for students to master in physics.

Acceleration

Acceleration is a measure of how quickly the velocity of an object is changing. It's a vector quantity, meaning it has both magnitude and direction. In SI units, acceleration is measured in meters per second squared (\(\text{m/s}^2\)).

If an object speeds up, slows down, or changes direction, it is accelerating. The direct relationship Newton's second law establishes between force and acceleration can be written as \( a = F / m \), meaning acceleration can be determined by dividing the force exerted on an object by its mass.

In our exercise, block \(m_{2}\) is experiencing an acceleration of \(a_{2} = 2.6 \; \text{m/s}^{2}\), which is given. With this information and the formula provided by Newton's second law, students can find the acceleration of other connected objects, such as \(m_{1}\), based on the force exerted through a spring or other means.

Mass

Mass is the measure of the amount of matter in an object. In the International System of Units (SI), mass is measured in kilograms (kg). Mass is a scalar quantity, which means it has magnitude but no direction, unlike force or acceleration.

An object's mass is a critical factor in force and motion calculations. Newton's second law, \( F = m \times a \), clearly indicates that the mass of an object affects the force required to change its motion. When dealing with problems involving multiple objects, such as the two blocks in our exercise, it is essential to consider the mass of each object separately.

The mass of an object remains constant, regardless of its location or the presence of a force. This is why we can use mass as a reliable quantity when solving problems related to motion, as seen in the calculation of the accelerations of \(m_{1}\) and \(m_{2}\) in our exercise.

Frictionless Motion

Frictionless motion is a concept where objects move without the opposing force of friction. This makes calculations much simpler because we can ignore the energy loss due to heat and the force that typically acts opposite to the direction of motion.

In a frictionless environment, like the one mentioned in our exercise with blocks sliding on a horizontal table, the only forces in play are those directly applied to the object, such as tension from springs or ropes. In reality, frictionless surfaces do not exist, but considering frictionless motion allows students to focus on and understand the fundamental principles of dynamics without additional complexity.

When calculating force and acceleration in a frictionless scenario, all the input force goes into changing the velocity of the object. This leads to a direct and proportional relationship between force and acceleration, which is perfectly encapsulated in Newton’s second law, allowing predictions of the outcome to be more straightforward and theoretical concepts to be conveyed with greater clarity.