Question

Consider a block of rock with a height of \(1 \mathrm{~m}\) and horizontal dimensions of \(2 \mathrm{~m}\). The density of the rock is \(2750 \mathrm{~kg} \mathrm{~m}^{-3}\). If the coefficient of friction is \(0.8,\) what force is required to push the rock on a horizontal surface?

Short answer

The required force is 86328 N.

Step-by-step solution

Determine the Volume of the Block

The block is rectangular with a height of 1 m and horizontal dimensions of 2 m. The volume, therefore, is given by:\[V = ext{height} \times ext{width} \times ext{length} = 1 \times 2 \times 2 = 4 \text{ m}^3\]

Calculate the Mass of the Block

Using the given density \(2750 \text{ kg/m}^3\), calculate the mass: \[m = V \times ext{density} = 4 \times 2750 = 11000 \text{ kg}.\]

Find the Weight of the Block

The weight \(W\) of an object on Earth's surface is the product of its mass \(m\) and the acceleration due to gravity \(g \approx 9.81 \text{ m/s}^2\):\[W = m \times g = 11000 \times 9.81 = 107910 \text{ N}.\]

Calculate the Frictional Force

The frictional force, which is the force required to push the block, is given by the equation for friction: \[F_{\text{friction}} = \mu \times W,\]where \(\mu = 0.8\) is the coefficient of friction.\[F_{\text{friction}} = 0.8 \times 107910 = 86328 \text{ N}.\]

Key concepts

Density of Rock

Understanding the density of rock is key to solving many physics problems involving force and weight. Density is defined as mass per unit volume and is typically expressed in units of kilograms per cubic meter (kg/m³). In this exercise, the density of the rock block is given as 2750 kg/m³. This means that every cubic meter of the rock has a mass of 2750 kilograms.
To calculate the mass of the rock block, you need to first determine its volume by measuring its dimensions. Here, the block's volume is found to be 4 cubic meters, as calculated by multiplying its height and horizontal dimensions: 1 m x 2 m x 2 m. Once you know the volume, you can easily calculate the mass by multiplying the density by the volume. Understanding density helps in transitioning to subsequent steps like mass calculation and force determination.

Coefficient of Friction

The coefficient of friction plays a crucial role in determining how much force is required to move an object across a surface. It measures the degree of resistance a surface provides to the movement of an object. The coefficient is dimensionless and varies based on the materials in contact. In this scenario, the coefficient of friction is 0.8, indicating a fairly significant degree of resistance between the rock and the horizontal surface.
The frictional force, which counteracts motion, is directly proportional to the weight of the object and the coefficient of friction. A higher coefficient means more force is required to overcome the friction. Calculating the frictional force involves multiplying this coefficient by the weight of the object, allowing us to find out how much force is needed to push the rock.

Acceleration due to Gravity

An essential factor in calculating weight and force, the acceleration due to gravity is a constant that quantifies the force of Earth's gravitational pull on objects. Represented as \( g \,\approx\, 9.81 \,\text{m/s}^2 \), it is used to calculate the weight of objects on Earth's surface by the formula: weight (\