The horizontal surface on which the objects slide is frictionless. If M = 2.0 kg, the tension in string 1 is 12 N. Determine F.
A)25 N
B)20 N
C)30 N
D)35 N
E)40 N

Question

The horizontal surface on which the objects slide is frictionless. If M = 2.0 kg, the tension in string 1 is 12 N. Determine F. 
A)25 N 
B)20 N 
C)30 N 
D)35 N 
E)40 N

Quizplus · Accepted Answer

Using Newton's second law, we know that the forces in the y-direction must balance each other, as the object is not accelerating vertically. Thus, we can find the tension in string 2:
T2 - M * g = 0
T2 - 2.0 kg * 9.8 m/s^2 = 0
T2 = 19.6 N

Next, we can use the fact that the acceleration of the system is the same for both masses. Thus, we can write another equation using Newton's second law in the x-direction:
F - T1 = M * a
F - 12 N = 2.0 kg * a

We can also write an equation for the acceleration by considering the tension forces:
T1 - T2 = (M1 + M2) * a
12 N - 19.6 N = (2.0 kg + M) * a
-7.6 N = (2.0 kg + M) * a

We can solve these two equations simultaneously to find F:
F - 12 N = 2.0 kg * a
F - 12 N = (2.0 kg / (2.0 kg + M)) * (-7.6 N)
F - 12 N = -3.8 N / (1 + 0.5M)

Using the given value for T1, we can solve for M:
12 N = T1 = T2 = (2.0 kg + M) * a
a = 12 N / (2.0 kg + M)

Substituting this into the previous equation:
F - 12 N = -3.8 N / (1 + 0.5M)
F = -3.8 N / (1 + 0.5M) + 12 N

Now we can substitute the known values to solve for F:
F = -3.8 N / (1 + 0.5(2.0 kg)) + 12 N
F = 20 N

The Horizontal Surface on Which the Objects Slide Is Frictionless