# Motion up an Inclined Plane - Not sure if I did it right

• jumbogala
In summary, to find the total time for a brick projected up an inclined plane with an initial speed v0 to return to its original position, you must first consider the x-axis to be along the inclined plane and calculate the two forces acting along it - the x component of gravity and the friction. By integrating the equation for acceleration and using the initial conditions, you can find the velocity of the brick at any time. To find the time it takes to travel up the incline, set the velocity to 0 and solve for time. To find the time for the brick to return to its original position, you will need to use the same method but also take into account the changing acceleration as the brick travels back down the incline. If
jumbogala

## Homework Statement

A brick is projected up an inclined plane with an initial speed v0.

If the inclination of the plane is 30 degrees and the coefficient of sliding friction μ = 0.1, find the total time for the block to return to its original position.

## The Attempt at a Solution

Consider the x-axis to be along the inclined plane. Then there are two forces acting along the x axis: the x component of gravity, and the friction.

X component of gravity: mgsin(30)
fricition: mgcos(30)μ, since mgcos(30) is the normal force on the brick

Then ma = -mgsin(30) - mgcos(30)μ or a = -gsin(30) - gcos(30)μ

a = dv/dt, so:

dv = (-gsin(30) - gcos(30)μ) dt

Integrating both sides gives

v = -gsin(30)t - gcos(30)μt + C, where C is a constant of integration

The initial conditions are at t = 0, v = v0 so C = V0

Then v = -4.905t - 0.85t + V0 = -5.755t + V0

Going up the incline, I set v = 0 and find that t = V0 / 5.755

I'm not sure if I can just double the above, because on the way up the incline gravity and friction are in the same direction. On the way down, they are in opposite directions.

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No, you can't double it for the reason you suspect. As it slides back down, it will cover the same distance, but its acceleration will be different, so it'll take more time getting back down.

So if I just to the same thing, but down the plane instead of up, then add the two times together, it should work.

It's a little different since you don't know the final velocity.

I didn't think of that...

I guess it would be better to find the distance the brick travels up the slope, then use that distance on the way back down.

So instead of using calculus to do this, on the way up I could find acceleration and distance.

Then on the way down, I could also find acceleration, and I'd have distance, so I could find time. I think, anyway.

How would this work if the force on the block wasn't constant though? The reason I wanted to use calculus is that in class we're dealing more with changing forces than static ones now.

In this problem, you have constant accelerations, so there's really no need to resort to calculus as you presumably have the kinematic equations for constant acceleration. What you did was essentially rederive one of those equations.

If the force isn't constant, you may need to use calculus to solve the equation. You'll see this when you study simple harmonic motion.

## 1. What is motion up an inclined plane?

Motion up an inclined plane refers to the movement of an object along a ramp or slope that is at an angle with the horizontal surface.

## 2. How is the motion affected by the incline of the plane?

The incline of the plane affects the motion by increasing or decreasing the force required to move the object, as well as changing the direction of the force.

## 3. What is the relationship between the angle of the incline and the force required?

The force required to move an object up an inclined plane is directly proportional to the angle of the incline. As the angle increases, the force required also increases.

## 4. How does the weight of the object affect its motion up an inclined plane?

The weight of the object plays a crucial role in motion up an inclined plane. The greater the weight, the more force is needed to overcome the force of gravity and move the object up the incline.

## 5. What factors can affect the motion up an inclined plane?

Aside from the angle and weight, factors such as friction, surface texture, and air resistance can also affect the motion up an inclined plane. These factors can either increase or decrease the force required to move the object.

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