Chapter 3: Q3.4-18E (page 116)

When an object slides on a surface, it encounters a resistance force called friction. This force has a magnitude of $μN$ , where $μ$ the coefficient of kinetic friction and N isthe magnitude of the normal force that the surface applies to the object. Suppose an object of mass30 kgis released from the top of an inclined plane that is inclined $30 °$ to the horizontal (see Figure 3.11). Assume the gravitational force is constant, air resistance is negligible, and the coefficient of kinetic friction $μ = 0.2$ . Determine the equation of motion for the object as it slides down the plane. If the top surface of the plane is 5 m long, what is the velocity of the object when it reaches the bottom?

Short Answer

Expert verified

The velocity of the object when it reaches the bottom is $V (t) = 5.66 m / \sec$ and $x (t) = 5.66 t + c$ .

Step by step solution

Find the value of velocity

There are two forces are written as,

$F_{1} = m g \sin 30^{o} F_{2} = - μ m g \cos 30^{o}$

Now put the given values then;

$m \frac{d v}{d t} = m g s i n 30^{o} - μ m g c o s 30^{o} \frac{d v}{d t} = g s i n 30^{o} - μ g c o s 30^{o} (Puttingthevaluesof g = 0.2) \frac{d v}{d t} = 3.207$

Since $v (t) = V (x (t))$ ,

So, the values are written as;

$\frac{d v}{d t} = V \frac{d V}{d x} V \frac{d V}{d x} = 3.207$

Find the value of velocity by limits.

Now, find the value of velocity then,

$\int V d V = \int_{0}^{5} 3.207 d x \frac{V^{2}}{2} - {3.207 x|}_{0}^{5} = 0 (Integratewithlimits 0 to 5) V^{2} = 32.07 V (t) = 5.66 m / \sec$

Evaluate the equation of motion.

Now, for the value of x(t),

$v = 5.66 \frac{d x}{d t} = 5.66 x (t) = 5.66 t + c$

Therefore, the results are $x (t) = 5.66 t + c$ and $v (t) = 5.66 m / \sec$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Find the value of velocity

Find the value of velocity by limits.

Evaluate the equation of motion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Discrete Mathematics

Applied Mathematics

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.