Okay, let's solve this step-by-step:
1. We have a cart with mass 60 kg and we want to find the maximum force P that can be applied without making either wheel reaction zero.
2. Let's label the wheel reactions as RA (at wheel A) and RB (at wheel B). According to Newton's second law, the sum of forces must equal mass times acceleration:
RA + RB + P = ma
3. For either RA or RB to be zero, the other reaction would have to equal the entire force P. So as a maximum, we have:
RB = P or RA = P
4. Let's assume RB = P is the maximum (the logic is the same for RA = P). Then plugging into Newton's second law:
P + P+ P = ma
3P = ma
P = ma/3
5. Since m = 60 kg, the maximum P that can be applied without a zero wheel reaction is:
P = (60 kg)(a)/3
6. Plugging in the knowns:
P = (60 kg)(a)/3
P = 20 kg * a
So the largest force P that can be applied is 20 kg times the acceleration a.
7. To find a, the acceleration, we use Newton's second law again with the total forces and mass:
RA + RB + P = ma
RA + RB = ma - P
RA + RB = ma - 20a
(RA + RB)/2 = m(a - 20a)/2
RA = m(a/2 - 10a)
Since RA must be greater than 0, the maximum a is 2 m/s2.
So in summary:
The maximum force P that can be applied is 20 kg x 2 m/s2 = 40N
The acceleration of the cart is 2 m/s2