Mass sliding down an incline plane(with friction)

[Benzoate]

Homework Statement

A mass slides on an incline plane with friction at an angle theta from the horizontal. The mass starts at rest at a height of h1 + h2 and then slides off a ramp at height h1.

a) If the coefficient of friction when a block is sliding down a plane is mu, what is the velocity vector of the mass when the mass is at height h1 ?(as the mass leaves the ramp?) Assume that the magnitude of the frictional force takes standard form F(frictional), N being the normal force exerted by the incline plane of the ramp

b) How far does the mass travel before it hits the ground?

Homework Equations

my_solution
a)

x component: mg*sin(theta)-F(friction)=m*a

y component: N-m*g*cos(theta)=0 ==> N= m*g*cos(theta)

mg*sin(theta)-mu*m*g*cos(theta) ==> a=g*sin(theta)-mu*g*cos(theta)

I now integrate a to obtain v and I have:

v= g*t*sin(theta)-mu*g*cos(theta)

from the figure, I can convert my sin(theta) into h2/(h2^2+x^2)^(1/2) and same thing with my cos(theta);

Therefore, v=g*t*h2/(h2^2+x^2)^(1/2)+ mu*g*x/(h2^2+x^2)^(1/2)

b) delta(x) is a little bit trickier to calculate.


x component: v(x)=v1*cos(theta)
F=m*dv(x)/dt = 0 since now that the mass is not supported by an inclined plane the only force acting on it is gravity.

delta(x)=(v1*cos(theta)*delta(t)

y component: F=m*dv(y)/dt=-m*g, v(y)=v(0y)-g*delta(t)
delta(y)=v(0y)*t-.5*g*(delta(t))^2


-h1=-(v1*sin(theta))*delta(t)-.5*g*t^2

(delta(t))^2+(2*v1*sin(theta)/g)*delta(t)-2*h1/g=0

using the quadratic formula , my delta(t) is:

delta(t)=(2*v1*sin(theta)/g) +- sqrt((2*v1*sin(theta)+4*(2*h1/g))/2

now delta(x)=v1*cos(theta)*delta(t) and I can easily plug in v1 from part a to calculate delta(x)

Is my solution correct?

The Attempt at a Solution

 

[Benzoate]

anyone not understand my question?

 

[alphysicist]

Hi Benzoate,

Homework Statement

A mass slides on an incline plane with friction at an angle theta from the horizontal. The mass starts at rest at a height of h1 + h2 and then slides off a ramp at height h1.

a) If the coefficient of friction when a block is sliding down a plane is mu, what is the velocity vector of the mass when the mass is at height h1 ?(as the mass leaves the ramp?) Assume that the magnitude of the frictional force takes standard form F(frictional), N being the normal force exerted by the incline plane of the ramp

b) How far does the mass travel before it hits the ground?

Homework Equations

my_solution
a)

x component: mg*sin(theta)-F(friction)=m*a

y component: N-m*g*cos(theta)=0 ==> N= m*g*cos(theta)

mg*sin(theta)-mu*m*g*cos(theta) ==> a=g*sin(theta)-mu*g*cos(theta)

I now integrate a to obtain v and I have:

v= g*t*sin(theta)-mu*g*cos(theta)

from the figure, I can convert my sin(theta) into h2/(h2^2+x^2)^(1/2) and same thing with my cos(theta);

Therefore, v=g*t*h2/(h2^2+x^2)^(1/2)+ mu*g*x/(h2^2+x^2)^(1/2)

It appears that you have time in your equation; I believe they would want the answer to only involve quantities which they mention in the problem statement.

The acceleration you found looks right; so how can you use that to find the velocity at the end of the ramp?

(Also, they mention they want the velocity vector, which I would think either means magnitude or direction, or in i-hat,j-hat form, depending on your instructor. But maybe you already had that in mind?)