If a rolling object has a radius of R and rolls with a speed v, what is its kine
Practice Questions
Q1
If a rolling object has a radius of R and rolls with a speed v, what is its kinetic energy?
(1/2)mv^2
(1/2)mv^2 + (1/2)Iω^2
(1/2)mv^2 + (1/2)(1/2)mR^2(v/R)^2
None of the above
Questions & Step-by-Step Solutions
If a rolling object has a radius of R and rolls with a speed v, what is its kinetic energy?
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 calculated using the formula TKE = (1/2)mv^2, where m is the mass of the object and v is its speed.
Step 3: The rotational kinetic energy (RKE) is calculated using the formula RKE = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
Step 4: For a solid cylinder or sphere, the moment of inertia I can be expressed as (1/2)mR^2, and the angular velocity ω can be related to the speed v by the formula ω = v/R.
Step 5: Substitute ω into the RKE formula: RKE = (1/2)(1/2)mR^2(v/R)^2 = (1/4)mv^2.
Step 6: Now, add the translational kinetic energy and the rotational kinetic energy together: Total KE = TKE + RKE = (1/2)mv^2 + (1/4)mv^2.
Step 7: Combine the terms: (1/2)mv^2 + (1/4)mv^2 = (3/4)mv^2.
Step 8: Therefore, the total kinetic energy of the rolling object is (3/4)mv^2.
Kinetic Energy of Rolling Objects – The total kinetic energy of a rolling object is the sum of its translational kinetic energy (1/2 mv^2) and its rotational kinetic energy (1/2 Iω^2), where I is the moment of inertia and ω is the angular velocity.
Moment of Inertia – For a solid cylinder or disk, the moment of inertia I is (1/2)mR^2, which is crucial for calculating rotational kinetic energy.
Relationship Between Linear and Angular Velocity – For rolling without slipping, the relationship v = Rω must be used to relate linear speed to angular speed.
