A wheel of radius R and mass M is rolling without slipping on a horizontal surfa
Practice Questions
Q1
A wheel of radius R and mass M is rolling without slipping on a horizontal surface. If it has a linear speed v, what is its total kinetic energy? (2022)
(1/2)Mv²
(1/2)Mv² + (1/2)(Iω²)
(1/2)Mv² + (1/2)(Mv²)
(1/2)Mv² + (1/2)(Mv²/2)
Questions & Step-by-Step Solutions
A wheel of radius R and mass M is rolling without slipping on a horizontal surface. If it has a linear speed v, what is its total kinetic energy? (2022)
Step 1: Understand that the total kinetic energy of the wheel is made up of two parts: translational kinetic energy and rotational kinetic energy.
Step 2: The formula for translational kinetic energy (K.E.) is (1/2)Mv², where M is the mass of the wheel and v is its linear speed.
Step 3: The formula for rotational kinetic energy is (1/2)(Iω²), where I is the moment of inertia and ω is the angular speed.
Step 4: For a solid cylinder (which is the shape of the wheel), the moment of inertia I is (1/2)MR², where R is the radius of the wheel.
Step 5: Since the wheel is rolling without slipping, the relationship between linear speed v and angular speed ω is given by ω = v/R.
Step 6: Substitute ω = v/R into the rotational kinetic energy formula: (1/2)(Iω²) becomes (1/2)((1/2)MR²)(v/R)².
Step 7: Simplify the rotational kinetic energy: (1/2)((1/2)MR²)(v²/R²) = (1/4)Mv².
Step 8: Now, add the translational and rotational kinetic energies together: Total K.E. = (1/2)Mv² + (1/4)Mv².
Step 9: Combine the terms: Total K.E. = (2/4)Mv² + (1/4)Mv² = (3/4)Mv².
Translational Kinetic Energy – The energy due to the linear motion of the center of mass of the wheel, calculated as (1/2)Mv².
Rotational Kinetic Energy – The energy due to the rotation of the wheel about its center of mass, calculated as (1/2)(Iω²), where I is the moment of inertia.
Moment of Inertia – A measure of an object's resistance to changes in its rotation, for a solid cylinder it is I = (1/2)MR².
Rolling Without Slipping – A condition where the point of contact between the wheel and the surface does not slide, linking linear and angular velocities (v = Rω).
