Find the tension in the string. Answer as an expression b. Find the period of the circular motion. Answer as an expression A mass m1 undergoes circular motion of radius R on a horizontal frictionless table, connected by a massless string through a hole in the table to a second mass m2(figure below). If m2 is stationary
The Correct Answer and Explanation is :
To solve the problem of finding the tension in the string and the period of the circular motion, we first need to understand the forces involved in this system. Let’s analyze each part step-by-step:
1. Tension in the String
The setup involves two masses, ( m_1 ) and ( m_2 ). Mass ( m_1 ) is moving in a circle on a horizontal frictionless table, and is connected by a string through a hole in the table to mass ( m_2 ), which hangs vertically below the table.
Since mass ( m_2 ) is stationary and only subject to gravitational and tension forces, the tension ( T ) in the string must balance the gravitational force on ( m_2 ). Therefore, the tension ( T ) is given by:
[ T = m_2 g ]
where ( g ) is the acceleration due to gravity.
2. Period of the Circular Motion
To find the period of the circular motion of mass ( m_1 ), we need to consider the centripetal force required to keep ( m_1 ) moving in a circle of radius ( R ). This centripetal force is provided by the tension in the string.
The centripetal force ( F_c ) required to keep ( m_1 ) in circular motion is:
[ F_c = \frac{m_1 v^2}{R} ]
where ( v ) is the speed of ( m_1 ).
From the balance of forces on ( m_2 ), we know ( T = m_2 g ). This tension also acts as the centripetal force for ( m_1 ), so:
[ m_2 g = \frac{m_1 v^2}{R} ]
From which we can solve for ( v ):
[ v = \sqrt{\frac{m_2 g R}{m_1}} ]
The period ( T ) of the motion is the time it takes for ( m_1 ) to make one complete revolution. The circumference of the circle is ( 2\pi R ), and with speed ( v ), the period ( T ) is:
[ T = \frac{2\pi R}{v} = \frac{2\pi R}{\sqrt{\frac{m_2 g R}{m_1}}} = 2\pi \sqrt{\frac{m_1 R}{m_2 g}} ]
Explanation
In this system, the tension in the string provides a crucial link between the two masses, balancing the gravitational pull on ( m_2 ) and providing the necessary centripetal force to keep ( m_1 ) in circular motion. The period of the motion is influenced by the mass ratio of ( m_1 ) to ( m_2 ), the radius of the circle, and the gravitational constant. A larger ( m_1 ) or smaller ( m_2 ) would result in a longer period, indicating that ( m_1 ) would move slower, making wider circles due to the weaker centripetal pull relative to its mass. Conversely, a larger ( m_2 ) results in a stronger tension, faster motion for ( m_1 ), and thus a shorter period.
This dynamics showcase fundamental principles of circular motion and the interplay of forces in a two-body system connected by a string, illustrating key concepts in classical mechanics.
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