## Physics Olympiads Series Problem 12

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- Written by fisikastudycenter

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**Problem 12**

A mechanical system consisting of four masses m_{1}, m_{2}, m_{3} and m_{4}. See the figure.

A horizontal force F is now applied to cart m_{4} as shown. The size of F is such that m_{1}, m_{2} and m_{3} remain at rest relative to m_{4}. Ignore the mass and frictional forces of string and pulley, find the magnitude of F and the system acceleration ! Use g = 10 m/s^{2} and the four masses respectively are 0,3 kg, 0,4 kg, 0,5 kg and 2,8 kg!

**Problem source:** Fisika Study Center-Modification of ^{3}rd International Physics Olympiad 1969, Brno, Czechoslovakia, Problem 1

**Solution**

Analyze the four masses as a system moving on x axis with accelerations a_{1} = a_{2} = a_{3} = a_{3} = a_{4} = a, Apply Newton's law

Analyze m_{3} at vertical direction, y axis, in equilibrium conditions.The acting forces on m_{3} are W = mg and te string tension, T_{23}.

Analyze m_{1} at x axis, horizontal direction, the acting force is the string tension T_{12}.

Analyze m_{2}, the acting forces are strin tensions T_{12} and T_{23}.

Combining equation 4 and 2 and 3, and then equation 1 to get the result.

**Try This**

Figure 1 shows a mechanical system consisting of three carts A, B and C of masses m_{1} = 0.3 kg, m_{2} =0.2 kg and m_{3} = 1.5 kg respectively. Carts B and A are connected by a light taut inelastic string which passes over a light smooth pulley attaches to the cart C as shown. For this problem, all resistive and frictional forces may be ignored as may the moments of inertia of the pulley and of the wheels of all three carts. Take the acceleration due to gravity g to be 9.81 m s^{−2}.

1. A horizontal force *F* is now applied to cart C as shown. The size of *F* is such that carts A and B remain at rest relative to cart C.

a) Find the tension in the string connecting carts A and B.

b) Determine the magnitude of *F*.

2. Later cart C is held stationary, while carts A and B are released from rest.

a) Determine the accelerations of carts A and B.

b) Calculate also the tension in the string.

Kunci:

1) 2.94 N dan 29.4 N

2) 5.89 m s^{−2} dan 1.18 N

**Sumber Soal: **^{3}rd International Physics Olympiad 1969, Brno, Czechoslovakia, Problem 1

Prepared bY : Fisika Study Center

**Sumber-sumber:**

-Soal International Physics Olympiad 1969, Brno, Czechoslovakia

-Mekanika 1, Yohanes Surya

-Marthen Kanginan, OSN SMA

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