A cylinder rolls down a hill. If it has a radius R and rolls without slipping, what is the relationship between its linear velocity v and its angular velocity ω?
Practice Questions
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Q1
A cylinder rolls down a hill. If it has a radius R and rolls without slipping, what is the relationship between its linear velocity v and its angular velocity ω?
v = Rω
v = 2Rω
v = ω/R
v = R^2ω
For rolling without slipping, the relationship is v = Rω.
Questions & Step-by-step Solutions
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Q
Q: A cylinder rolls down a hill. If it has a radius R and rolls without slipping, what is the relationship between its linear velocity v and its angular velocity ω?
Solution: For rolling without slipping, the relationship is v = Rω.
Steps: 6
Step 1: Understand that a cylinder rolling down a hill means it is moving forward while also spinning.
Step 2: Identify the two types of velocities: linear velocity (v) which is how fast the center of the cylinder is moving, and angular velocity (ω) which is how fast the cylinder is spinning around its axis.
Step 3: Recognize that when the cylinder rolls without slipping, the point of contact with the ground does not slide. This means that the distance the cylinder rolls forward is equal to the distance it spins.
Step 4: Relate the linear distance traveled by the cylinder to its rotation. The distance traveled by the cylinder in one complete rotation is equal to the circumference of the cylinder, which is 2πR (where R is the radius).
Step 5: Set up the relationship: When the cylinder rolls one full turn, it moves forward a distance of 2πR. This means that the linear distance (d) is equal to the radius (R) times the number of rotations (n).
Step 6: The linear velocity (v) is the distance traveled over time, and the angular velocity (ω) is the number of rotations per unit time. Therefore, we can express this relationship as v = Rω.
