Help with power and force problems

[pimuni]

1. A ski lift carries skiers along a 600-meter slope inclined at 30 degrees. Each chair has a mass of 50 kg, and each rider has a mass of 70 kg. The skier gets off at the top of the hill, so the chairs are empty for the return trip. Under maximum load conditions six riders per minute arrive at the top. If 60% of the energy supplied by the motors goes into overcoming friction, what average power must the motor supply?

This is what I have so far, but I am not sure if it is right:

W = F*d*cos(theta)
Power = W/t

since it says 6 ppl per minute make it to the top, i will use 6 ppl times their weight to get the force required. 6*70*9.8 = 4116 N
since the change in height is 300 m (600*sin30(theta)), then W = 4116(300) = 1236060 J
it said 6 ppl in 1 minute, do that's 60 seconds
power = W / t, so 1236060 j / 60 seconds = 20601 watts
since this is only 40% of the energy, 100% will be 51502.5 watts

im not sure if I did this right or not, because i didn't account for the mass of the chairs, since half are going up and half r going down, so their work cancels.

2 .
(53 and 27 are angles)
_______________
\ 53....37/
...\.../
T1 \.../ T2
...\.../
...[x]

T1 = .15 m
T2 = .2 m

A ball weight 5 Newtons is suspended by two strings as shown. Suppose the ball swings as a pendulum perpendicular to the plane of the page, achieving a maximum speed of 0.6 meters per second during its motion. Determine the magnitude and direction of the net force on the ball as it swings through the lowest point in its path.

here i am a bit stuck. i know that the lowest point is where the maximum speed will be held at. But I am not sure if the different angles make the net force point somewhere else than south. The horizontal acceleration will be 0, and the vertical will be 9.8. What else could I infere?

 

[berkeman]

Welcome to the PF. One of the rules here is that you must show your work in order to get help on homework problems. Show us what you have done so far on each problem, and where you are stuck.

 

[kyle8921]

To solve this problem, we can use the equation F=ma, where F is the net force, m is the mass of the ball, and a is the acceleration. Since the ball is swinging perpendicular to the plane of the page, the acceleration will only be in the vertical direction.

First, we need to find the mass of the ball. We know that the weight of the ball is 5 Newtons, and we can use the formula F=mg to find the mass. Therefore, m = F/g = 5/9.8 = 0.51 kg.

Next, we can use the equation for centripetal force, F=mv^2/r, to find the magnitude of the net force at the lowest point. Here, v is the maximum speed of 0.6 m/s and r is the distance from the ball to the center of rotation, which is equal to T1 + T2 = 0.15 + 0.2 = 0.35 m.

Substituting these values into the equation, we get F = (0.51)(0.6)^2/0.35 = 0.52 N.

Since the ball is moving towards the center of rotation at the lowest point, the direction of the net force will be towards the center, which is towards the south direction. Therefore, the net force on the ball at the lowest point is 0.52 N towards the south.

It is important to note that the angles of the strings do not affect the net force on the ball, as the tension in the strings will balance out and cancel each other out. The only forces acting on the ball are its weight and the centripetal force, which is the net force.