Tension problems kill my soul. Please assist?

Question

Find the tension in each cable supporting the 540 N cat burglar in Figure P4.15. Assume the angle of the inclined cable is 36.0°.

http://www.webassign.net/sf5/p4_11alt.gif

inclined cable:
horizontal cable:




and also:

A van accelerates down a hill (Fig. P4.71), going from rest to 30.0 m/s in 7.10 s. During the acceleration, a toy (m = 0.350 kg) hangs by a string from the van's ceiling. The acceleration is such that the string remains perpendicular to the ceiling.
http://www.webassign.net/sf/p4_71.gif
angle = 25.5
find the tension of the string:


=D Thanks much!

Answer 1

Since the cable to the left is perfectly horizontal, all of the vertical force must be counter-acted by the vertical component of tension in the cable:

T*sin(36)=540
T=540/sin(36)

The horizontal must counteract the left and right the same
540/cos(36) is the horizontal

for the second, the tension in the string should be
m*g*cos(25.5)

check
9.81*sin(25.5)=4.22 m/s^2

30=a*7.1
a=4.22
yes, it checks

How did I know that?

Look at a FBD of the toy

Perpendicular to the ceiling, the forces are
m*g*cos(25.5)=T
horizontal
m*g*sin(25.5)=m*a
or
a=g*sin(25.5)



j

Answer 2

The first one is "statics" and shouldn't be too soul-wrenching. It's a matter of finding the components of all the forces and making sure that the forces add up to zero in each direction. So, you've got weight straight down countered by the upward parts of each cable force, and the horizontal cable bits need to be equal and opposite. At its most complex, this will give you as many equations as there are cables, which becomes an algebra problem (how complex depends on how many cables, but for two cables it's not too hard).

The second one is a bit trickier, since the acceleration isn't zero. Here's the key: you need to tilt your axes so that going along the road is your x-axis, and "perpendicular to the ceiling" is the y-axis. To keep the string along the y-axis, you need to be in "freefall" along the x-axis...the acceleration must equal the acceleration of gravity along the x-axis. It turns out that the problem supplies enough information that you don't need to work out the acceleration at all! If all the givens actually fit together, you just need the tension to counter the y-axis component of weight, and that's it. The x-axis component of weight will be "countered" by the fact you're in x-direction freefall.