Review. A thin, cylindrical rod ℓ=24.0 cm long with mass m=1.20 kg has a ball of diameter d=8.00

step 1 In this question, we have a broad and a ball combination. Initially, they are vertically, they a

Review. A thin, cylindrical rod ℓ=24.0 cm long with mass...

Review. A thin, cylindrical rod $\ell=24.0 \mathrm{cm}$ long with mass $m=1.20 \mathrm{kg}$ has a ball of diameter $d=8.00 \mathrm{cm}$ and mass $M=2.00 \mathrm{kg}$ attached to one end. The arrangement is originally vertical and stationary, with the ball at the top as shown in Figure $\mathrm{P} 10.50 .$ The combination is free to pivot about the bottom end of the rod after being given a slight nudge. (a) After the combination rotates through 90 degrees, what is its rotational kinetic energy? (b) What is the angular speed of the rod and ball? (c) What is the linear speed of the center of mass of the ball? (d) How does it compare with the speed had the ball fallen freely through the same distance of 28 $\mathrm{cm}$ ?

Key Concepts

Free-Fall Kinematics

Free-fall kinematics deals with the motion of objects under the influence of gravity alone, where acceleration is constant. The linear speed of an object after falling through a given height can be determined using kinematic equations or energy considerations, and this provides a useful reference to compare with speeds attained by objects in rotational motion.

Linear Speed in Rotational Systems

In a rotating system, there is a relationship between the angular velocity and the linear speed of a point at a particular distance from the rotation axis. This relationship is given by v = r ?, where v is the linear speed, r is the distance from the axis, and ? is the angular velocity. This concept helps in determining the translational speed of parts of a rotating object.

Angular Velocity

Angular velocity measures the rate at which an object rotates and is usually defined in radians per second. It directly influences the rotational kinetic energy of a system. This concept is particularly important when converting energy changes—resulting from gravitational potential energy loss—into rotational motion.

Conservation of Energy

In rotational dynamics, the conservation of energy principle states that the total mechanical energy (potential plus kinetic) remains constant in the absence of non-conservative forces like friction. This principle is used to relate changes in gravitational potential energy to gains in kinetic energy, whether the energy is translational or rotational.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation about an axis. It is given by the formula K = 0.5 I ?², where I is the moment of inertia and ? is the angular velocity. This concept links the energy in a rotating system to its geometry and speed of rotation.

Moment of Inertia

The moment of inertia is a measure of how mass is distributed about an axis and how resistant an object is to changes in its rotational motion. It plays a role analogous to mass in linear motion. When dealing with composite objects, like a rod with an attached sphere, the total moment of inertia is determined by summing the moments of inertia of each component (calculated based on the shape, mass, and distance from the axis of rotation).

Transcript

00:01 In this question, we have a broad and a ball combination.

00:05 Initially, they are vertically, they are vertically as shown, and then it's given a slight nudge, and then it's going to, the whole combination is going to rotate through 90 degrees.

00:17 And there are four parts in this question in part a.

00:20 We want to find out the rotational kinetic energy of the combination after it rotates through 90 degrees.

00:31 So we'll be using conservation of energy.

00:42 Okay, on the ball, rod, earth system.

00:53 Okay, so there's no external force acting on this system.

00:57 So the energy will be conserved.

00:59 So ki plus ui, the total energy of the system will be conserved.

01:04 Okay.

01:06 And then it goes to kf.

01:07 Plus uf okay the initial kinetic energy is zero the initial gravitational potential energy is zero the final kinetic energy will be rotational kinetic energy okay rotational and then for the final gravitational potential energy so we need to visualize the final position of the objects you'll find out how much distance that they have trouble so the rod, it has fallen through the distance of l over 2 by looking at the center of the rod.

01:52 And then for the ball, it has fallen through the distance of l plus d over 2.

02:00 So k rotation is equal to, rotation of kinetic energy is equal to mg, small m times g, l2 plus big m times g, l plus d over 2.

02:13 So we can put in the numbers.

02:16 So the 9 .8 is the common term and then small m is 1 .2 l is 24 cm, a comma 2 meters you get 0 .24.

02:32 The big m is 2k and then 0 .24 plus 0 .08 0 by 2.

02:41 Calculate you get 6 .90 juice.

02:44 Okay, so this is the answer for part a.

02:49 Next, in part b, we need to calculate the angular speed of the rod and the ball at this instance.

02:57 So first we need to find the total moment of inertia...

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