# What is the minimum force ? What is the minimum force F must be applied to lift the body mass m? The height of the inclined plane is equal to H and the length of the inclined plane is equal to L. Consider the blocks to be weightless and neglect friction.

###### Solution. Posted on Saturday, December 27, 2014.

If a movable pulley moved a distance l than the force F performed work equal to F*l. The body on the inclined plane at the same time moved a distance 2*l and rose to a height of h. The amount of gravitational potential energy (m*g*h) possessed by an elevated the body is equal to the work done. Thereby, m*g*h=F*l. The h could be found from the similarity of triangles.
h = 2*l*H/L
Now, when looking at the previous equation, it can be written as m*g*2*l*H/L=F*l
Finally,
F=2*m*g*H/L

## 3 thoughts on “What is the minimum force ?”

1. Johnny B says:

First find the angle, theta, between the plane
and floor.

theta = arcsin(H/L)

The force, f, against the rope parallel
to the incline is

f = m*g*cos(theta)

which is the force needed to lift it up
the incline. The pulley at the end of the
incline doesn’t multiply the force f, it
just transmits it to one side of the other
pulley.

The minimum force, F, to lift the mass, m,
is twice the two forces pulling up of the
pulley pulling down. The two forces lifting
up are equal to f.

F = 2*m*g*cos(theta).

It’s hard to answer this problem with mostly
words, but fun.

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• Boris Sapozhnikov says:

Thank you for your solution but

the Answer is F = 2mgH/L

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• Johnny B says:

I see, looking at it again. It’s the sin, not the cosine,
and sin(arcsin(H/L)) gives H/L. Thanks.

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