# What Is the Coefficient of Kinetic Friction for a Block on an Inclined Plane?

• EchoTheCat
In summary: So, 0.75 is the answer.In summary, using the given information of a block with a mass of 820 g starting from rest at the top of an inclined plane that is 106 cm long and makes an angle of 49.5 degrees with the horizontal, and reaching the bottom in 0.89 seconds, the coefficient of kinetic friction on the ramp is 0.75. This was calculated using the equations x=vot +0.5at2 and uk = Fk/N, where a = 2.67642 m/s/s, Fk = 2.195 N, and Fn = 5.219 N.
EchoTheCat

## Homework Statement

A block with a mass of 820 g starts from rest at the top of an inclined plane that is 106 cm long and makes an angle of 49.5 degrees with the horizontal. It accelerates uniformly down the ramp and reaches the bottom in 0.89 s. What is the coefficient of kinetic friction on the ramp?

x=vot +0.5at2
uk = Fk/N

## The Attempt at a Solution

In order to calculate coefficient of friction, I need the normal force and Fk.
vo = 0 (starts at rest)
x = 106 cm = 1.06 m
t = 0.89 sec
a = ?
1.06 = 0.5(0.89 2)(a)
a = 2.67643 m/s/s
Then, Fk = (2.67643 m/s/s)(0.82 kg) = 2.1947 N
N = (0.82 kg) (9.8 m/s/s) cos 49.5 degrees = 5.219 N
2.1947/5.219 = 0.421

EchoTheCat said:
a = 2.67643 m/s/s
Then, Fk = (2.67643 m/s/s)(0.82 kg) = 2.1947 N

You're going to want to rethink this. The net force was friction? Where is gravity in all of this?

Draw a diagram and show all the forces on the block. That is always the correct starting point.

RedDelicious said:
You're going to want to rethink this. The net force was friction? Where is gravity in all of this?
The vertical component of gravity is 0.82*9.8*cos 49.5 degrees = 5.219
The horizontal component of gravity would be 0.82*9.8*sin 49.5 degrees = 6.1106.
So, Fx(net) = 6.1106 - 2.1947 = 3.9159 ?
And uf = 3.9159/5.219 = 0.750 ?

EchoTheCat said:
The vertical component of gravity is 0.82*9.8*cos 49.5 degrees = 5.219
The horizontal component of gravity would be 0.82*9.8*sin 49.5 degrees = 6.1106.
So, Fx(net) = 6.1106 - 2.1947 = 3.9159 ?
And uf = 3.9159/5.219 = 0.750 ?

You perhaps mean the normal and tangential components of gravity. Think this through:

You calculated the acceleration.

What can you work out immediately from this? Hint: think of Newton's laws.

PeroK said:
You perhaps mean the normal and tangential components of gravity. Think this through:

You calculated the acceleration.

What can you work out immediately from this? Hint: think of Newton's laws.
I don't think I understand what you're saying.

Mass is given as 0.82 kg, and I calculated acceleration to be 2.67642. So then force would be 2.195.

As PeroK says: You should always start from first principles. If you draw the diagram, you will know what are all the forces acting on the object. Then you can write Newton's law. Then you can define a coordinate system, take components, and then look for the unknowns.
Incidentally, your use of the words "vertical" and "horizontal" is not corect. You should refer to a coordinate system which you define< in this case, as an axis parallel to the inclined plane and one perpendicular to the inclined plane (again, diagram!).

theta = 49.5 degrees
Fg = (0.82 kg) (9.8 m/s/s) = 8.036 N
Fn = Fgy = Fg cos 49.5 degrees = 8.306*cos 49.5 degrees = 5.219 N
Fa = 2.67643 m/s/s * 0.82 kg = 2.195 N
Fgx = Fg * sin 49.5 degrees = 6.1106 N

#### Attachments

• force diagram.PNG
7.6 KB · Views: 485
Good. So, what is Newton's second law?

net force = mass x acceleration

EchoTheCat said:
net force = mass x acceleration

Okay, so first you calculated the acceleration. From that you got the total net force.

Now, what forces are acting on the block? In fact, you have those in your diagram. So, which force do you need to calculate now?

I need to calculate Fk, the force of kinetic friction.

EchoTheCat said:
I need to calculate Fk, the force of kinetic friction.

So, how do you do that? Think about what forces you already know.

Wouldn't Fk = Fgx - Fa = 6.1106 - 2.195 = 3.9156 N?

EchoTheCat said:
Wouldn't Fk = Fgx - Fa = 6.1106 - 2.195 = 3.9156 N?

If ##F_a## is the net force, then that is correct.

Finally, how do you get the coefficient of friction from the frictional force?

3.9159/5.219 = 0.750

EchoTheCat said:
3.9159/5.219 = 0.750

Well, those are three numbers. But what are they?

Fk = 3.9159
Fn = 5.219
uf = Fk/Fn = 3.9159/5.219 = 0.750

EchoTheCat said:
Fk = 3.9159
Fn = 5.219
uf = Fk/Fn = 3.9159/5.219 = 0.750

The method looks right. I haven't checked the numbers, though.

PS ##\mu = 0.75## looks right!

You can only really give the answer to two decimal places.

Last edited:

## 1. How does the angle of the ramp affect the speed of the block sliding down?

The steeper the angle of the ramp, the faster the block will slide down. This is because the force of gravity acting on the block is greater when the ramp is at a steeper angle, causing it to accelerate more quickly.

## 2. What is the relationship between the mass of the block and its acceleration down the ramp?

According to Newton's Second Law of Motion, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Therefore, the greater the mass of the block, the slower it will accelerate down the ramp.

## 3. How does friction affect the motion of the block down the ramp?

Friction is a force that opposes motion, so it will act in the opposite direction of the block's movement down the ramp. This means that friction will slow down the block's acceleration and reduce its speed as it slides down the ramp.

## 4. What is the minimum angle of the ramp that the block can slide down without any external force?

In order for the block to slide down the ramp without any external force, the ramp must be at an angle of 0 degrees. This means that the ramp is completely horizontal and there is no component of gravity acting on the block in the direction of motion.

## 5. How can the potential energy and kinetic energy of the block be calculated as it slides down the ramp?

The potential energy of the block can be calculated using the equation PE = mgh, where m is the mass of the block, g is the acceleration due to gravity, and h is the height of the ramp. As the block slides down, this potential energy is converted into kinetic energy, which can be calculated using the equation KE = 1/2 mv^2, where m is the mass of the block and v is its velocity.

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