Solving Calculus Problems After Many Years - Sheldon

[SheldonG]

I am reviewing my calculus (it has been many years). Hope this is ok to ask, amongst all the young'ins...

I was doing fine until:

Homework Statement

1) A mass M is drawn up a straight incline of given height h by a mass m which is attached to the first mass by a string passing from it over a pulley at the top of the incline and which hangs vertically. Find the angle of the incline in order that the time of ascent be a minimum.

2) A swinging pendulum is 4 feet long is rotating at the rate of 18 deg/sec when it makes an angle of 30 with the vertical. How fast is the end of the pendulum rising or falling at that moment.

Homework Equations

F=ma, trig functions

The Attempt at a Solution

For 1, I calculated the net force on M as Ma = 32m - 32 M sin A. Reasoning that the greatest acceleration would also make the least time, I do:

a = 32m/M - 32 sin A.

I just treat this as a derivative (which it is), and set it to zero, solving for sin A:

sin A = m/M

However, the book (Morris Kline's calculus) gives sin A = m/2M.

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For 2, the pendulum is sweeping out a circle, so s = rA, where a is the angle with the vertical. Differentiating, I get ds/dt = r dA/dt. From the problem data, r = 4 ft, dA/dt = 18 deg/sec = pi /10 rad/sec. ds/dt is v, the velocity, so the answer should be 4 (pi/10) = 2*pi/5 ft/sec. But the answer in the book is pi/5 ft/sec.

I thought perhaps they might want the vertical speed. For that I get vy = v/sin 30 = 2v = 4*pi/5 ft/sec. Even farther away.

Any help you can give to help this old guy keep his brain alive would be appreciated.

Thank you,
Sheldon

 

[jing]

Problem 1

Some thoughts to help you.

You need to check what you have written about the net forces to find a, it is not correct.

There is a tension T in the string connecting the two masses. Consider the forces on each mass separately obtain two equations both containing T then add them to eliminate T

You then have an equation for the acceleration a.

If the mass M is to go up an incline of height h how far will this mass actually travel?

Assuming an initial velocity of zero use

s=ut+(1/2)t^2

to get a relationship between the angle A and the time taken t.

You can then find dA/dt to look for turning points


PS I'm no youngster

 

[SheldonG]

Thank you, Jing. I found my error, thanks to you. I really appreciate your help.

I also solved the second problem, thanks to your stimulus, by using h = l-l*cos(A).

Your help is very appreciated.

Sheldon

 

[TristanH]

I am having trouble with the fist problem here as well.

The equation

F = 32m - 32M sin A

(where F is the net force on M) is given in the book. I know also that F=ma, yet this equation only accounts for one mass. I also know I can find a maximum of something by setting its derivative to 0, yet I'm not sure what derivative to set to zero, since none of the information given involves time. Overall, I'm not sure where to begin to tackle this one. Help!