Solution to problem 13.7.10 from the collection of Kepe O.E.

13.7.10 In this problem, we consider a tripod with a mathematical pendulum, which moves down an inclined plane with acceleration a = g sin?. It is necessary to determine the angle ? at which the ball is in a position of relative rest, if the angle ? equals 10°. The answer to the problem is 0.

To solve this problem it is necessary to use the law of conservation of energy. Initially, the kinetic energy of the system is zero, so the potential energy must be zero at any time. The potential energy of the system is calculated by the formula Ep = mgh, where m is the mass of the ball, g is the acceleration of free fall, h is the height of the ball above the zero level.

It can be seen from the figure that there is no friction force between the ball and the plane, therefore the work of gravity is equal to the work of the normal reaction force of the plane. Work of gravity W1 = mgh sin?, where h = l(1 - cos?), where l is the length of the thread of the mathematical pendulum. Work done by the normal reaction force W2 = -mgcos?l.

From the law of conservation of energy it follows that the work done by gravity must be equal to the work done by the normal reaction force: mgh sin? = -mgcos?l.

From here we can express the angle ? at which the ball is in a position of relative rest: tg? = sin?/cos? = -l/h = -1/(1 - cos10°) ≈ -6.88. Corner ? in this case it is equal to 0.

Solution to problem 13.7.10 from the collection of Kepe O.?.

We present to your attention the solution to problem 13.7.10 from the collection of Kepe O.?. This digital product contains a detailed solution to the problem using the law of conservation of energy. The problem considers a tripod with a mathematical pendulum, which moves down an inclined plane with acceleration a = g sin?.

To solve the problem, it is necessary to determine the angle ? at which the ball is in a position of relative rest, if the angle ? equals 10°.

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Solution to problem 13.7.10 from the collection of Kepe O.?. is associated with determining the angle of inclination of the plane along which a tripod with a mathematical pendulum moves, provided that the angle of inclination of the plane is 10 degrees and the tripod moves downward with an acceleration equal to the acceleration of gravity multiplied by the sine of the angle of inclination of the plane. To solve the problem it is necessary to use the laws of mechanics and kinematics formulas.

In this problem you need to find the angle? in a position of relative rest of the ball. To do this, you can use the law of conservation of energy, according to which the potential energy of the body at the initial point is equal to the kinetic energy of the body at the final point.

Thus, the following equation can be written:

mgh = (1/2)mv^2

where m is the mass of the ball, g is the acceleration of gravity, h is the height of the starting point, v is the speed of the ball at the end point.

The height of the starting point is zero, since the ball is in a position of relative rest. The speed of the ball at the end point can be found using the kinematics formula:

v^2 = u^2 + 2as

where u is the initial velocity equal to zero, a is the acceleration of the ball along the inclined plane, s is the distance traveled by the ball.

The distance traveled by the ball can be found using the following formula:

s = l (1 - cos ?)

where l is the length of the mathematical pendulum, ? - plane inclination angle.

Thus, the following equation can be written:

mgh = (1/2)ml^2(?')^2 + (1/2)ml^2(g sin ?)cos ?

Where ?' - angular velocity of a mathematical pendulum.

To solve this equation it is necessary to express the angle? through known quantities and solve the resulting equation. As a result of solving the equation, it turns out that the angle? equal to zero.

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