Deriving a formula for KE (rolling + projection)

In summary, the process of deriving a formula for kinetic energy (KE) when considering both rolling and projection involves understanding the contributions of translational and rotational motion. The total kinetic energy is the sum of translational kinetic energy (1/2 mv²) and rotational kinetic energy (1/2 Iω²), where m is mass, v is linear velocity, I is the moment of inertia, and ω is angular velocity. By applying the relationship between linear and angular velocities (v = rω, where r is the radius), the formulas can be combined to express the total kinetic energy of an object in motion that is both rolling and projected.
  • #1
foreverlostinclass
1
0
Homework Statement
A ball is rolled down a ramp, then projected a distance R from the end of a curved ramp. The kinetic energy of the ball when it lands is given by the equation: E_k=(gmR^2)/4h, where g is the gravitational acceleration constant (9.81 m/s^2), m is the mass of the ball & h is the height from the ground to the bottom of the ramp.
Edit: Actually it's the kinetic energy when the ball is projected, not when it lands (sorry, misread the description).
Relevant Equations
moment of inertia of a solid sphere: I = 2/5mr^2
K = 1/2mv^2 + 1/2Iw^2
U = mgh
I'm not sure where the equation E_k=(gmR^2)/4h comes from & I also don't really know where to start either :(
 
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  • #2
According to our rules, to receive help, you need to show some credible effort towards answering the question. How about telling us what you do know and how you would approach this problem?
Please read, understand and follow our homework guidelines, especially item 4, here
https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/

Also, it would help if you posted the statement of the problem as was given to you. Note that you are saying that the kinetic energy of the ball is given at the moment of projection but it is not at all clear what you are asked to find.
 
  • #3
foreverlostinclass said:
The kinetic energy of the ball when it lands is given by the equation: E_k=(gmR^2)/4h, where g is the gravitational acceleration constant (9.81 m/s^2), m is the mass of the ball & h is the height from the ground to the bottom of the ramp.
Edit: Actually it's the kinetic energy when the ball is projected, not when it lands (sorry, misread the description).
Relevant Equations: moment of inertia of a solid sphere: I = 2/5mr^2
K = 1/2mv^2 + 1/2Iw^2
U = mgh
It is still not quite true. That formula ignores the rotational KE.
To deduce it, you have to assume the bottom of the ramp is horizontal.
Find
  • how long it would take to land, in terms of g and h
  • the relationship between that time, R, and the velocity with which it leaves the ramp.
 

FAQ: Deriving a formula for KE (rolling + projection)

What is kinetic energy (KE) and how is it calculated?

Kinetic energy (KE) is the energy that an object possesses due to its motion. It is calculated using the formula KE = 1/2 mv², where m is the mass of the object and v is its velocity. For rolling objects, we also consider rotational kinetic energy.

How do you derive the formula for kinetic energy of a rolling object?

The total kinetic energy of a rolling object is the sum of its translational kinetic energy and its rotational kinetic energy. The translational kinetic energy is given by 1/2 mv², while the rotational kinetic energy is given by 1/2 Iω², where I is the moment of inertia and ω is the angular velocity. For a rolling object, ω can be expressed as v/r, where r is the radius. Thus, the total KE for a rolling object is KE_total = 1/2 mv² + 1/2 I(v/r)².

What is the moment of inertia, and why is it important in calculating KE?

The moment of inertia (I) is a measure of an object's resistance to rotational motion about an axis. It depends on the mass distribution of the object relative to that axis. In calculating the kinetic energy of a rolling object, the moment of inertia is crucial because it determines how much energy is associated with the object's rotation, which contributes to its total kinetic energy.

How does the radius of a rolling object affect its kinetic energy?

The radius of a rolling object affects its rotational kinetic energy through the relationship between linear velocity and angular velocity (ω = v/r). A larger radius typically means that for a given linear velocity, the angular velocity will be smaller, which affects the distribution of kinetic energy between translational and rotational forms. However, the total kinetic energy is independent of the radius, as it is a function of mass and velocity.

Can you explain the difference between translational and rotational kinetic energy?

Translational kinetic energy refers to the energy due to the linear motion of an object and is calculated using the formula KE_trans = 1/2 mv². Rotational kinetic energy, on the other hand, is associated with the object's rotation around an axis and is calculated using the formula KE_rot = 1/2 Iω². For rolling objects, both forms of kinetic energy contribute to the total kinetic energy, which must be considered when analyzing motion.

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