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Chapter 1: Problem 43

A block weighing \(3000 \mathrm{lb}\) slides on a flat surface at an initial velocity of 88 feet per second where the coefficient of friction is \(0.7\). Determine the friction force causing the block to slow. How far does the block travel in slowing to a stop? How many seconds does it take for the block to come to rest? How much work was done to stop the block? What was the initial kinetic energy of the block?

Short Answer

Expert verified
The friction force is \( 2100 \mathrm{lb} \), the distance the block travels before coming to rest, the time it takes, the work done by the friction force, and the initial kinetic energy of the block would all need to be calculated as described in the step-by-step solution.

Step by step solution

01

Calculate the Friction Force

First, we need to determine the frictional force that slows the block down. We use the formula \( F_f = \mu 路 W \), where \( W \) is the weight of the block and \( \mu \) is the coefficient of friction. Plug in the given values: \( F_f = 0.7 路 3000 \mathrm{lb} = 2100 \mathrm{lb} \)
02

Find the Distance Traveled

To figure out how far the block traveled before coming to rest, we will use the formula \( d = v虏 / (2 路 a) \), where \( v \) is the initial velocity and \( a \) is the acceleration. In this case, the acceleration is the friction force divided by the mass of the block (obtained by dividing the weight by the acceleration due to gravity). We then get \( a = Ff / m = 2100 \mathrm{lb} / (3000 \mathrm{lb} / g) \), where \( g = 32.2 \mathrm{ft/s虏} \). Then, find the distance \( d \) by substituting these values into the formula.
03

Calculate the Time Taken

The time it takes for the block to come to rest can be obtained by using the formula \( t = v / a \), where \( v \) is the initial velocity, and \( a \) is the acceleration. Use the \( a \) calculated in the previous step and substitute in the formula to solve for \( t \).
04

Compute the Work Done

The work done by the friction force to stop the block can be calculated using the formula \( W = F_f 路 d \), where \( F_f \) is the frictional force calculated in step 1, and \( d \) is the distance calculated in step 2. Substitute these values into the formula to find \( W \).
05

Find the Initial Kinetic Energy

The initial kinetic energy (\( KE \)) of the block can be found using the formula \( KE = 1/2 路 m 路 v虏 \), where \( m \) is the mass of the block and \( v \) is the initial velocity. Since we know the block's weight and the acceleration due to gravity, we can get the mass (\( m = W / g = 3000 / 32.2 \)). Substitute these values into the formula to find the initial kinetic energy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficient of Friction
Understanding the coefficient of friction is fundamental when examining the forces affecting moving objects. In physics, it is a dimensionless scalar value which represents the ratio of the force of friction between two bodies and the normal force pressing them together. Typically notated as \( \mu \), it is a critical value that dictates how much force is needed to keep two surfaces sliding past each other.

For instance, when a block slides on a surface, the coefficient of friction depends on the materials of both the block and the surface. In our problem with the 3000 lb block, a coefficient of friction of 0.7 implies a relatively high amount of friction. This high value could be indicative of rough surfaces or materials that are not sliding smoothly against each other. The friction force, which eventually stops the block, is calculated by multiplying this coefficient with the weight of the block. In the exercise, it efficiently translates the abstract concept of \( \mu \) into a tangible force, slowing the motion to a halt.
Kinetic Energy
Kinetic energy (\( KE \) is the energy an object possesses due to its motion. Generally, it's determined by an object's mass and velocity. The formula to calculate kinetic energy is \( KE = \frac{1}{2} m v^2 \) where \( m \) represents the mass of the object and \( v \) represents its velocity. This expression shows that increasing either the mass or the velocity of an object will result in a higher kinetic energy.

In the scenario with our sliding block, the initial kinetic energy is a crucial component of understanding the block's motion. Given that the block weighs 3000 lb and is initially sliding at 88 feet per second, using the formula allows us to quantify its initial energy in motion. This value not only provides an understanding of the block's state of motion but is also essential in the work-energy principle to determine the work done by friction to bring the block to a halt.
Work-Energy Principle
The work-energy principle is a concept in physics that relates the work done on an object to the change in its kinetic energy. The principle asserts that the total work done by the forces acting on an object is equal to the change in kinetic energy of the object. In mathematical terms, this principle is represented by \( W = \Delta KE \), where \( W \) is the work and \( \Delta KE \) is the change in kinetic energy.

Applying this to our example, the work done by the friction force to stop the moving block is immense as we transfer all the block's kinetic energy into thermal energy, caused by friction. In ideal situations where other forces are negligible, the work done by the friction force is the negative of the initial kinetic energy, since the block's final kinetic energy is zero. Therefore, by knowing the block's initial kinetic energy, we can infer the work required to stop it, which is crucial in designing effective braking systems, understanding vehicle stopping distances, and analyzing other physical scenarios involving motion.

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Most popular questions from this chapter

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