Studdy AI Logo
Math  /  Algebra

Question(II) A train locomotive is pulling two cars of the same mass behind it, Fig. 4-43. Determine the ratio of the tension in the coupling (think of it as a cord) between the locomotive and the first car (FT11)\left(F_{\mathrm{T1} 1}\right) to that between the first car and the second car (FT2)\left(F_{\mathrm{T} 2}\right), for any nonzero acceleration of the train.
FIGURE 4-43 Problem 37.

Studdy Solution

STEP 1

1. The train locomotive is pulling two cars of the same mass.
2. The tension in the coupling between the locomotive and the first car is FT1 F_{\mathrm{T1}} .
3. The tension in the coupling between the first car and the second car is FT2 F_{\mathrm{T2}} .
4. The train is accelerating with a nonzero acceleration a a .
5. The mass of each car is m m .

STEP 2

1. Analyze the forces acting on the second car.
2. Analyze the forces acting on the first car.
3. Determine the expressions for FT1 F_{\mathrm{T1}} and FT2 F_{\mathrm{T2}} .
4. Calculate the ratio FT1FT2 \frac{F_{\mathrm{T1}}}{F_{\mathrm{T2}}} .

STEP 3

Analyze the forces acting on the second car. The only horizontal force acting on the second car is the tension FT2 F_{\mathrm{T2}} . According to Newton's second law, the net force is equal to the mass times the acceleration:
FT2=m⋅a F_{\mathrm{T2}} = m \cdot a

STEP 4

Analyze the forces acting on the first car. The first car is being pulled by the tension FT1 F_{\mathrm{T1}} and is pulling the second car with tension FT2 F_{\mathrm{T2}} . The net force on the first car is:
FT1−FT2=m⋅a F_{\mathrm{T1}} - F_{\mathrm{T2}} = m \cdot a

STEP 5

Determine the expressions for FT1 F_{\mathrm{T1}} and FT2 F_{\mathrm{T2}} . From Step 1, we have:
FT2=m⋅a F_{\mathrm{T2}} = m \cdot a
From Step 2, solve for FT1 F_{\mathrm{T1}} :
FT1=m⋅a+FT2 F_{\mathrm{T1}} = m \cdot a + F_{\mathrm{T2}}
Substitute FT2=m⋅a F_{\mathrm{T2}} = m \cdot a into the equation:
FT1=m⋅a+m⋅a=2m⋅a F_{\mathrm{T1}} = m \cdot a + m \cdot a = 2m \cdot a

STEP 6

Calculate the ratio FT1FT2 \frac{F_{\mathrm{T1}}}{F_{\mathrm{T2}}} :
FT1FT2=2m⋅am⋅a=2 \frac{F_{\mathrm{T1}}}{F_{\mathrm{T2}}} = \frac{2m \cdot a}{m \cdot a} = 2
The ratio of the tension in the coupling between the locomotive and the first car to that between the first car and the second car is:
2 \boxed{2}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord