AP Physics 1:: Torque Rotation

What is TORQUE, and what is the mathematical FORMULA relating torque to an applied FORCE and the LEVER ARM (perpendicular distance from the rotation axis to the force's line of action)?

- **A)** Torque depends only on the MAGNITUDE of the applied force, with no actual dependence on distance from the rotation axis
- **B)** Torque is calculated as tau = F/r (force divided by distance), rather than force multiplied by distance (and the angle factor)
- **C)** This concept has no actual mathematical relationship between torque, an applied force, and the lever arm distance
- **D)** Torque is the ROTATIONAL analog of force -- it measures a force's tendency to cause ROTATION about an axis; tau = F*r*sin(theta), where r is the distance from the axis to the point of force application and theta is the angle between the force and the position vector

Why does applying a force FARTHER from the rotation axis (a LONGER lever arm) produce a LARGER torque than applying the SAME force CLOSER to the rotation axis, based on the torque formula tau = F*r*sin(theta)?

- **A)** Torque is actually INVERSELY proportional to lever arm distance, meaning a SHORTER lever arm would produce a LARGER torque -- the reverse of the actual relationship
- **B)** Since torque is DIRECTLY PROPORTIONAL to the lever arm distance r (for a given force and angle), increasing the distance from the rotation axis directly increases the resulting torque -- this is why using a longer wrench or door handle makes rotating/opening easier with the same applied force
- **C)** This concept has no actual relationship between lever arm distance and the resulting torque produced by a given force
- **D)** Lever arm distance affects only the DIRECTION of the resulting torque, not its magnitude

Why does a force applied EXACTLY ALONG the line connecting the point of application to the rotation axis (i.e., theta = 0 or 180 degrees) produce ZERO TORQUE, based on the formula tau = F*r*sin(theta)?

- **A)** Torque depends only on the force's magnitude and distance from the axis, with no actual dependence on the angle between them
- **B)** A force applied along this line always produces the MAXIMUM possible torque, rather than zero torque
- **C)** Since sin(0 degrees) = sin(180 degrees) = 0, a force applied directly along the line to the rotation axis (with NO perpendicular component) produces ZERO torque, regardless of the force's magnitude or the distance from the axis
- **D)** This concept has no actual relationship between a force's angle relative to the rotation axis and whether it produces any torque

What is ROTATIONAL INERTIA (also called MOMENT OF INERTIA), and what does it represent about an object's RESISTANCE to changes in its ROTATIONAL motion?

- **A)** Rotational inertia depends only on an object's total mass, with no actual dependence on how that mass is distributed relative to the rotation axis
- **B)** This concept has no actual relationship between rotational inertia and an object's resistance to changes in rotational motion
- **C)** Rotational inertia is the ROTATIONAL analog of mass -- it measures an object's resistance to ANGULAR acceleration (changes in rotational motion), and depends on BOTH the object's mass AND how that mass is DISTRIBUTED relative to the rotation axis
- **D)** Rotational inertia is identical to LINEAR inertia (mass), with no actual distinction between the two concepts for a rotating object

Why does an object with its mass distributed FARTHER from the rotation axis have a LARGER rotational inertia than an object of the SAME total mass with its mass distributed CLOSER to the axis (e.g., a hoop versus a solid disk of equal mass and radius)?

- **A)** Mass distributed FARTHER from the axis contributes LESS to rotational inertia than mass distributed closer to the axis, the reverse of the actual relationship
- **B)** Rotational inertia depends on mass multiplied by the SQUARE of its distance from the rotation axis (I = sum of m*r^2 for point masses) -- mass located farther from the axis contributes DISPROPORTIONATELY more to the total rotational inertia due to this squared distance dependence
- **C)** Rotational inertia depends only on TOTAL mass, with no actual dependence on how far that mass is distributed from the rotation axis
- **D)** This concept has no actual relationship between mass distribution and an object's rotational inertia

What is NEWTON'S SECOND LAW FOR ROTATION, and what is its mathematical FORMULA relating NET TORQUE to ROTATIONAL INERTIA and ANGULAR ACCELERATION?

- **A)** Newton's second law for rotation states that net torque equals rotational inertia multiplied by angular acceleration: tau_net = I*alpha -- this is the direct rotational analog of the familiar F_net = m*a from linear dynamics
- **B)** Newton's second law for rotation states that net torque equals rotational inertia DIVIDED BY angular acceleration, rather than multiplied by it
- **C)** Newton's second law for rotation applies only to objects with ZERO rotational inertia, making it inapplicable to real, physical rotating objects
- **D)** This law has no actual mathematical relationship between net torque, rotational inertia, and angular acceleration

What is ANGULAR VELOCITY, and how does it relate to an object's LINEAR (tangential) velocity for a point at a distance r from the rotation axis?

- **A)** Angular velocity (omega) measures the RATE of change of angular position (in radians per second); the relationship to linear (tangential) velocity is v = omega*r -- points FARTHER from the rotation axis move with a LARGER linear speed for the SAME angular velocity
- **B)** Angular velocity is identical to linear velocity, with no actual distinction or relationship requiring the radius r
- **C)** Linear velocity is calculated as v = omega/r (angular velocity divided by radius), rather than multiplied by radius
- **D)** This concept has no actual mathematical relationship between angular velocity and linear (tangential) velocity for a rotating object

What is ANGULAR ACCELERATION, and what is its mathematical RELATIONSHIP to LINEAR (tangential) ACCELERATION for a point at distance r from the rotation axis?

- **A)** This concept has no actual mathematical relationship between angular acceleration and tangential linear acceleration for a rotating object
- **B)** Angular acceleration is identical to linear acceleration, with no actual distinction or relationship requiring the radius r
- **C)** Angular acceleration (alpha) measures the RATE of change of angular velocity (in radians per second squared); the relationship to tangential linear acceleration is a_tangential = alpha*r -- directly analogous to the v=omega*r relationship for velocity
- **D)** Tangential linear acceleration is calculated as alpha/r (angular acceleration divided by radius), rather than multiplied by radius