A gymnast with mass m 1 = 46 kg is on a balance beam that sits on (but is not at

Question

A gymnast with mass m1 = 46 kg is on a balance beam that sits on (but is not attached to) two supports. The beam has a mass m2 = 117 kg and length L = 5 m. Each support is 1/3 of the way from each end. Initially the gymnast stands at the left end of the beam.

1)

What is the force the left support exerts on the beam?
N

2)

What is the force the right support exerts on the beam?
N

3)

How much extra mass could the gymnast hold before the beam begins to tip?
kg

4)

Now the gymnast (not holding any additional mass) walks directly above the right support.

What is the force the left support exerts on the beam?
N

5)

What is the force the right support exerts on the beam?
N

6)

At what location does the gymnast need to stand to maximize the force on the right support?

at the center of the beam

at the right support

at the right edge of the beam

Explanation / Answer

Given data is

Solution::--

forces upward = forces downward
(S1) + (S2) = (m1)(g) + (m2)(g)
(S1) + (S2) = (46 kg)(9.81 m/s²) + (117 kg)(9.81 m/s²)
(S1) + (S2) = 1599.03 N -===============> [1]

torques balance (using the right end as a reference):
(S1)(2/3)(L) + (S2)(1/3)(L) = (m1)(g)(L) + (m2)(g)(1/2)(L)
(S1)(2) + (S2) = 3[(m1)+ (m2)/2](g)
(S1)(2) + (S2) = 3[(46 kg)+ (117 kg)/2](9.81 m/s²)
(S1)(2) + (S2) = 3075.43 N ==========> [2]


1) )What is the force the left support exerts on the beam?

[2] - [1]:
(S1) = 3075.43 – 1599.03 N
(S1) = 1476.40 N

2) What is the force the right support exerts on the beam?
1476.40 N + (S2) = 1599.03 N
(S2) = 122.63 N

3) How much extra mass could the gymnast hold before the beam begins to tip?
Using the left support as the pivot,
(M)(g)(1/3)(L) = (m2)(g)(L/2 - L/3)
M = (m2)/2
M = (117 kg)/2
M = 58.5 kg

58.5 - 41 kg = 12.5 kg

4) Now the gymnast (not holding any additional mass) walks directly above the right support.

What is the force the left support exerts on the beam?

(117 x 9.8) = 1,147.77 N.
(46 x 9.8) = 451.26 N.
It is not important where the supports are placed, as without the gymnast on the beam, the weight is evenly distributed, so long as both supports are the same distance from the ends.
Weight on each support = (1,147.77/2) = 573.88 N.
It remains 573.88 N.

5) What is the force the right support exerts on the beam?

(573.88 + 451.26) = 1025.14 N.

6) At what location does the gymnast need to stand to maximize the force on the right support?
at the center of the beam
at the right support
at the right edge of the beam Yes

As the gymnast walks from the left of the beam to the right end of the beam, force on the right support increased from 573.88 N to 1025.14 N. I believe that this is maximum force.

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