# Incline problem with friction; alternative solution

In summary, the conversation discusses a problem in a textbook where a simple incline-mass problem (friction included) is solved using work-energy. The problem involves finding the speed of a child at the bottom of a slide given the mass, length of the slide, incline angle, and coefficient of kinetic friction. The solution is obtained using the equation W_{ext}=\Delta E_{mech}+f_k \Delta x and the final velocity is found to be approximately 5.6 m/s. However, the person discussing the problem attempted to solve it using Newton's second law and kinematic equations and obtained a significantly different answer of about 17.6 m/s. It is later realized that there was a mistake in

## Homework Statement

There is an example problem in a textbook I'm looking at where they solve a simple incline-mass problem (friction included) using work-energy.

We are given the mass 40kg, the length of the slide is 8 m, the incline is 30 degrees, the coefficient of kinetic friction is .35. We are looking for the child's speed at the bottom of the slide.

## Homework Equations

$$W_{ext}=\Delta E_{mech}+f_k \Delta x$$

Newton's Seconds Law, the basic kinematic equations:

## The Attempt at a Solution

The solution in the example is straight forward. The child-slide-earth system has no exterior forces acting so the equation I give above is set to zero.

$$0=mg\Delta h +\frac{1}{2}mv_f^2+f_k \Delta x=-mg\Delta x sin(\theta)+\frac{1}{2}mv_f^2 +mg\mu_k cos(\theta)\Delta x$$

Plugging in numbers and solving for the final velocity they obtain:
$$v_f \approx 5.6 m/s$$

I am attempting the same problem usuing Newton's second law and kinematic equations, but I am not obtaining a similar answer and I can't discern why not:

In the x-direction, using the usual coordinate system (x - axis parallels to slope of the incline, +x is down the incline), the acceleration should be:

$$a_x = gsin(\theta) -g \mu_k cos(\theta)$$

Plugigng in numbers real fast gives:
$$a_x \approx 1.93 m/s^2$$

Now I attempt to use kinematics to find the speed at the bottom of the slide:

$$x=\frac{1}{2}a_x t^2$$

$$v_f=a_x t$$

$$80m = \frac{1}{2}(1.93 m/s^2)t^2\Rightarrow t\approx 9.11s$$

Using this in the velocity equation along with the acceleration of 1.93 m/s^2 gives about 17.6 m/s for the speed. This is far different than the 5.6 m/s obtained using the energy equations.The numbers I obtained don't pass sanity check either.

In your last equation you've used 80 m as the length of the slide. Wasn't the value only 8 m in the problem statement?

gneill said:
In your last equation you've used 80 m as the length of the slide. Wasn't the value only 8 m in the problem statement?

You got to be kidding me. I can't believe I didn't notice that and wasted this much time. I guess I'm kind of relieved that it was my vigilance and not my understanding of physics that failed though. I thought I was going crazy for moment.

Last edited:

## 1. What is the incline problem with friction?

The incline problem with friction is a common physics problem that involves a block or object sliding down a ramp or incline with some amount of friction acting upon it. The goal is to calculate the acceleration, velocity, and distance traveled by the object.

## 2. What is the traditional solution to the incline problem with friction?

The traditional solution to the incline problem with friction involves using Newton's laws of motion and the concept of forces to calculate the acceleration and velocity of the object. This method requires the use of trigonometry to break down the forces into their components along the incline and perpendicular to it.

## 3. What is the alternative solution to the incline problem with friction?

The alternative solution to the incline problem with friction involves using conservation of energy instead of forces and acceleration. This method is often easier and quicker to use, as it does not require trigonometry calculations. However, it can only be used for certain scenarios where there is no change in the object's mass or height.

## 4. How do you determine the coefficient of friction in the incline problem?

The coefficient of friction can be determined by conducting experiments or by using data provided in the problem. It is a unitless value that represents the roughness or slipperiness of the surfaces in contact. It can also be calculated by dividing the frictional force by the normal force acting on the object.

## 5. Can the incline problem with friction be applied to real-life situations?

Yes, the incline problem with friction is a common physics problem that can be applied to real-life situations. For example, it can be used to calculate the stopping distance of a car on a sloped road or the speed at which a skier will reach the bottom of a ski slope. Understanding this problem can also help engineers design structures or machines that involve inclined surfaces.

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