Mechanics

[Enigma]

Could someone please help me with this question? I've had a go, but I don't really understand it.

 

A block of mass 3kg is at rest on a rough horizontal plane.

 

The block is acted on by a horizontal force of magnitude 14.7N. Given that the block is on the point of sliding, find the coefficient of friction between the block and the plane.

Friction = coefficient of friction x normal reaction

 

14.7 = coefficient x (3 x 9.8)

14.7 = coefficient x 29.4

 

Coefficient of friction = 0.5

 

The horizontal force is now replaced by a force of magnitude P N acting downards at 30º to the horizontal. Given that the block is again on the point of sliding, find the value of P.

 

 

The red is my working out - I'm not sure if it is correct. I don't know how to do the last part, so please could someone point me in the right direction.

 

Many thanks,

Enigma

[swansont]

If the force is now acting at an angle, there is an additional force down, meaning the normal force will be larger, since the acceleration is still zero. You will have to break P into its components.

[Enigma]

I have broken up P into its components, but I don't know where to go from there.

 

Horizontal = P cos30

Vertical = P sin30

 

Does P cos30 = 14.7?

 

Thanks,

Enigma

[Enigma]

After some thought:

 

R = 29.4 + P sin30

P cos30 = 0.5R

 

P cos30/0.5 = R

Substitute into top equation

 

P cos30/0.5 = 29.4 + P sin30

Multiply right-hand-side to get rid of fracion

 

P cos30 = 14.7 + 0.5(P sin30)

P cos30 = 14.7 + 0.5p + 0.25

P cos30 - 0.5p = 14.95

P(cos30 - 0.5) = 14.95

P = 14.95/(cos30 - 0.5)

 

P = 40.8N

 

Is that correct?

 

Thanks,

Enigma

[mezarashi]

Pluggin in P = 40.8N doesn't seem to satisfy a zero net force condition. The question says on the verge of slipping, so in the x-direction, Force-friction and horizontal component of P are equal. Friction can only act parallel to a surface, normal forces perpendicular.

 

Your freebody analysis should include the x and y axis.

 

x-axis

Pcos30 = Ffriction

 

y-axis

Psin30 + Fnormal = mg

 

But we know that

Ffriction = u x Fnormal, Fnormal = mg - Psin30

 

Try playing with those equations. Your answer shouldn't be more than 15N I'd suppose.

[Enigma]

OKay, I get this:

 

P cos30 = 0.5R

R = 29.4 - P sin30

 

P cos30 = 0.5(29.4 - P sin30)

P cos 30 - 14.7 - 0.5P sin30

P cos 30 + 0.5P sin30 = 14.7

P(cos30 + 0.5P sin30) = 14.7

 

P = 13.2N

 

But I don't understand how you got the equations out!

[mezarashi]

The equations are from the equilibrium condition. For example you know Newton's law:

 

Fnet = ma

 

If there is no acceleration (a), then there must be no net force. So we use this on the two axises, x and y.

 

In the y direction, the forces are, the reaction force (Fnormal), gravitational force (mg), and the y-component of P (Psin30).

 

In the x direction, again there is no acceleration so the force summation should be zero. The forces here are, Frictional Force (Ffriction) and the x-component of P (Pcos30). Take into consideration the directions and you will have the equations.

[Enigma]

Okay, I think I understand. I just hope I don't get a really hard question in the exam. Thank you very much

[Enigma]

I've managed to answer 3 parts of the following question, now I'm just stuck on the last part. Please can someone tell me how to go about this. Thanks.

 

The displacement of a particle at time t seconds after it passes through a fixed point is s m, where s = 4.8t + 0.06t^2 - 0.004t^3

 

Write down expressions in terms of t for

a) the velocity

v = 4.8 + 0.12t - 0.012t^2

 

b) the acceleration

a = 0.12 - 0.024t

 

c) Find the value of s when the acceleration of the particle is zero.

0 = 0.12 - 0.024t

0.12/0.024 = t

t = 5

s = (4.8 x 5) + (0.06 x 5^2) - (0.004 x 5^3) = 25.4

 

d) Find the distance travelled by the particle from the point where it reaches its maximum velocity to the point where its velocity is half its initial velocity.

 

I'm thinking that when the particle has reached its maximum velocity, the acceleration will be zero, so the time will be 5 seconds, but I don't know how to finish the question.

 

Many thanks,

Enigma

[swansont]

I've managed to answer 3 parts of the following question' date=' now I'm just stuck on the last part. Please can someone tell me how to go about this. Thanks.

 

The displacement of a particle at time t seconds after it passes through a fixed point is s m, where s = 4.8t + 0.06t^2 - 0.004t^3

 

Write down expressions in terms of t for

a) the velocity

[i']v = 4.8 + 0.12t - 0.012t^2[/i]

 

b) the acceleration

a = 0.12 - 0.024t

 

c) Find the value of s when the acceleration of the particle is zero.

0 = 0.12 - 0.024t

0.12/0.024 = t

t = 5

s = (4.8 x 5) + (0.06 x 5^2) - (0.004 x 5^3) = 25.4

 

d) Find the distance travelled by the particle from the point where it reaches its maximum velocity to the point where its velocity is half its initial velocity.

 

I'm thinking that when the particle has reached its maximum velocity, the acceleration will be zero, so the time will be 5 seconds, but I don't know how to finish the question.

 

Many thanks,

Enigma

 

 

You have an equation for v, so you can find the time when "its velocity is half its initial velocity." Now you have two times, and an equation that relates displacement and time.

[Enigma]

So can I find the initial velocity by putting zero into the velocity equation, and then find the time by using the initial velocity in the equation?

[mezarashi]

Yes, the velocity is at its maximum when the acceleration is zero. So you know that the maximum velocity occurs at t=5. A couple of other things after this. I'm assuming "distance travelled" really means what it means and not displacement from s=0. If that is the case.

 

1. Find the roots of the velocity equation and see if it the velecity reversed over the period t=0 to t=5.

2. If there are reversals, integrate the velocity in segments. Say from t=0 to t=3, then from t=3 to t=5. Take the absolute value of the answers and add them up (integrating velocity over time will give you distance).

 

As for the second part of the question, initial velocity = 4.8, half of it is 2.4, so find the value for t and do the same test as stated above.