If a rolling object has a mass m and radius r, what is the expression for its total kinetic energy?
Practice Questions
1 question
Q1
If a rolling object has a mass m and radius r, what is the expression for its total kinetic energy?
(1/2)mv^2
(1/2)mv^2 + (1/2)Iω^2
(1/2)mv^2 + (1/2)mr^2ω^2
(1/2)mv^2 + (1/2)(2/5)mr^2(ω^2)
The total kinetic energy of a rolling object is the sum of translational and rotational kinetic energy, which can be expressed as (1/2)mv^2 + (1/2)(2/5)mr^2(ω^2).
Questions & Step-by-step Solutions
1 item
Q
Q: If a rolling object has a mass m and radius r, what is the expression for its total kinetic energy?
Solution: The total kinetic energy of a rolling object is the sum of translational and rotational kinetic energy, which can be expressed as (1/2)mv^2 + (1/2)(2/5)mr^2(ω^2).
Steps: 7
Step 1: Understand that a rolling object has two types of motion: it moves forward (translational motion) and it spins (rotational motion).
Step 2: The translational kinetic energy (TKE) is given by the formula (1/2)mv^2, where m is the mass and v is the linear velocity of the object.
Step 3: The rotational kinetic energy (RKE) is given by the formula (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
Step 4: For a solid sphere or cylinder, the moment of inertia I can be expressed as (2/5)mr^2, where r is the radius of the object.
Step 5: Substitute the moment of inertia into the rotational kinetic energy formula: RKE = (1/2)(2/5)mr^2(ω^2).
Step 6: Combine the translational and rotational kinetic energy to find the total kinetic energy: Total KE = TKE + RKE.
Step 7: Write the final expression for total kinetic energy: Total KE = (1/2)mv^2 + (1/2)(2/5)mr^2(ω^2).
