In Fig. 6-45, a 1.34 kg ball is connected by means of two...

In Fig. 6-45, a $1.34 \mathrm{~kg}$ ball is connected by means of two massless strings, each of length $L=1.70 \mathrm{~m}$, to a vertical, rotating rod. The strings are tied to the rod with separation $d=1.70 \mathrm{~m}$ and are taut. The tension in the upper string is $35 \mathrm{~N}$. What are the (a) tension in the lower string, (b) magnitude of the net force $\vec{F}_{\text {net }}$ on the ball, and (c) speed of the ball? (d) What is the direction of $\vec{F}_{\text {net }} ?$

Key Concepts

Tension and Force Transmission

Tension is the pulling force transmitted along a string, cable, or rope when it is subjected to forces at its ends. Understanding how tension is distributed in systems such as multiple strings attached to a rotating body is crucial for analyzing how the forces combine to produce net acceleration and maintain motion.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path at a constant speed. It acts along the radius of the circle toward the center of rotation and is key to determining the object's angular speed and the configuration of other forces, such as tension, in rotating systems.

Rotational Dynamics

Rotational dynamics involves the study of forces and torques associated with objects in rotation. It encompasses applying Newton’s laws to situations where the forces not only produce linear accelerations but also result in circular motion, which is particularly evident in systems where multiple forces (like tensions) balance to provide the necessary centripetal force.

Force Decomposition and Equilibrium

Force decomposition involves breaking down forces into their components, typically along perpendicular axes, to simplify the analysis of their combined effects. In systems with strings and rotational motion, decomposing the tensions into vertical and horizontal components allows us to determine the net forces and verify equilibrium conditions in the vertical direction while ensuring that the horizontal components provide the centripetal force needed for circular motion.

Transcript

00:01 For this problem on the topic of force and motion, we are shown in the figure a ball of mass 1 .34 kilograms that is connected by two massless strings, each of length 1 .7 meters to a vertical rod that is rotating.

00:14 The strings are tied to the rod with a separation of 1 .7 meters and are taught.

00:19 If we are given the tension in the upper string to be 35 newtons, we want to find the tension in the lower string, the magnitude of the net force on the ball, the speed of the ball, and the direction of this net force.

00:31 Now the free body diagram is shown here with the t u the tension exerted by the upper string on the ball, and tl the tension in the lower string.

00:42 M is the mass of the ball, and we can know that the tension in the upper string is greater than the tension in the lower string.

00:48 So it has to balance the downward pull of gravity and the force of the lower string.

00:53 So we'll take the positive x direction to be leftward toward the center of the circular path and positive y upward.

01:00 And we know that the magnitude of acceleration is v -colm, squared over r and from the and from newton's second law we have the x component of the forces t u cosine theta plus tl cosine theta is responsible for a centipital force and so this is mv squared over r where v is the speed of the ball and r the radius of its orbit the y component gives us t u theta minus tl sine theta minus the weight of the ball mg is equal to zero.

01:46 And so the second equation gives the tension in the lower string.

01:51 So the tension in the lower string, tl, is equal to tu minus mg over sine theta.

02:09 Now since the triangle is equilateral, this angle theta is 30 degrees...

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