Question

A small block of mass \(4 \mathrm{~kg}\) is placed inside a cart of mass 2 kg placed on horizontal surface. A force \(10 \mathrm{~N}\) is applied on the cart as shown. Neglect friction everywhere. The acceleration of the cart with respect to the ground is:(1) \(5 / 3 \mathrm{~m} / \mathrm{s}^{2}\) (2) \(2.5 \mathrm{~m} / \mathrm{s}^{2}\) (3) \(5 \mathrm{~m} / \mathrm{s}^{2}\) (4) \(1 \mathrm{~m} / \mathrm{s}^{2}\)

Short answer

The acceleration of the cart with respect to the ground is \( \frac{5}{3} \text{ m} / \text{s}^{2} \).

Step-by-step solution

- Understand the problem

A small block of mass 4 kg is placed inside a cart of mass 2 kg. A force of 10 N is applied to the cart. The goal is to find the acceleration of the cart with respect to the ground, without considering friction.

- Combine the masses

The total mass of the system (cart + block) can be calculated by adding the mass of the cart (2 kg) and the mass of the block (4 kg). Total mass: \[ m_{\text{total}} = m_{\text{cart}} + m_{\text{block}} = 2 \text{ kg} + 4 \text{ kg} = 6 \text{ kg} \]

- Apply Newton's Second Law

Newton's Second Law states that the force applied is equal to the product of the total mass and the acceleration of the system. \[ F = m_{\text{total}} \times a \]

- Solve for acceleration

Rearrange the equation to solve for acceleration (a): \[ a = \frac{F}{m_{\text{total}}} \]Plug in the values: \[ a = \frac{10 \text{ N}}{6 \text{ kg}} = \frac{10}{6} = \frac{5}{3} \text{ m} / \text{s}^{2} \]

Key concepts

acceleration calculation

Acceleration tells us how quickly an object changes its velocity. To calculate acceleration, we need to know the force applied to an object and its mass. According to Newton's Second Law, the acceleration \(a\) of an object is found using the equation: \[ a = \frac{F}{m} \] where \(F\) is the force and \(m\) is the mass. For example, in the problem above, the total force applied is 10 N, and the total mass of the system (cart + block) is 6 kg. Plugging these values into the formula, we get: \[ a = \frac{10 \, \text{N}}{6 \, \text{kg}} = \frac{5}{3} \, \text{m/s}^2 \] Therefore, the cart accelerates at \(\frac{5}{3}\, \text{m/s}^2\). Each unit of force moves the mass, causing its velocity to increase by this acceleration rate per second.

force and mass relationship

Newton's Second Law explains the relationship between force, mass, and acceleration. The law can be expressed using the formula: \[ F = m \cdot a \] This tells us that the force applied to an object is equal to the mass of the object multiplied by its acceleration. When multiple objects are combined, such as the 4 kg block and 2 kg cart, their masses add up to form a total mass. For this exercise, the combined mass is 6 kg. By blending meshes:

• The system's mass is increased, impacting acceleration if the force remains constant.
• The same force applied to a larger mass results in slower acceleration.

Due to this relationship, even though a 10 N force is applied, the greater combined mass of 6 kg results in an acceleration of \(\frac{5}{3}\, \text{m/s}^2\).

physics problem solving

Solving physics problems often involves applying formulas and understanding concepts like Newton's Laws. Here's a concise guideline to tackle similar problems:

• Understand the problem: Identify the given values and what you need to find.
• Inventory: Write down the masses and forces involved.
• Combine Forces: When required, sum the masses or forces.
• Apply Newton’s Second Law: Utilize \(F = m \cdot a\) to figure out the unknowns.
• Rearrange Equations: Solve for the unknown variable, like acceleration.

For our exercise:

1. Identified 4 kg and 2 kg as the masses of the block and cart, respectively.
2. Total mass was summed to 6 kg.
3. Force (10 N) applied to the combined mass provided the acceleration using \(a = \frac{F}{m} \).

Simplifying principles and practicing these steps can make solving physics problems straightforward and less intimidating. Always remember: practice makes perfect!