The point masses m and 2m lie along the x-axis, with m at the origin and 2m at x = L. A third point mass M is moved along the x-axis. At what point is the net gravitational force on M due to the other two masses equal to zero?

검증된 단계별 안내

Identify the forces acting on the mass M: The gravitational force on M due to mass m at the origin and the gravitational force on M due to mass 2m at x = L.

Write the expression for the gravitational force between two point masses using Newton's law of universal gravitation: \( F = \frac{G m_1 m_2}{r^2} \), where G is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and r is the distance between them.

Set up the equation for the net gravitational force on M to be zero: The force due to mass m at the origin should be equal in magnitude and opposite in direction to the force due to mass 2m at x = L.

Express the forces mathematically: \( \frac{G m M}{x^2} = \frac{G (2m) M}{(L-x)^2} \), where x is the distance of M from the origin.

Solve the equation \( \frac{1}{x^2} = \frac{2}{(L-x)^2} \) to find the position x where the net gravitational force on M is zero. This involves cross-multiplying and solving the resulting quadratic equation for x.

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주요 개념

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Gravitational Force

Gravitational force is the attractive force between two masses. It is described by Newton's law of universal gravitation, which states that the force is proportional to the product of the two masses and inversely proportional to the square of the distance between their centers. The formula is F = G * (m1 * m2) / r^2, where G is the gravitational constant.

Superposition Principle

The superposition principle states that the net force acting on a point mass is the vector sum of all individual forces acting on it. In the context of gravitational forces, this means that the total gravitational force on a mass is the sum of the gravitational forces exerted by each of the other masses independently.

Equilibrium Point

An equilibrium point is a position where the net force acting on an object is zero. For gravitational forces, this occurs when the gravitational pull from surrounding masses cancels out. In this problem, it involves finding a point along the x-axis where the gravitational forces from the two given masses on the third mass M are equal in magnitude but opposite in direction.