Curved and Rotational Motion

Question 1

Uniform Circular Motion

Answer 1

Object’s speed around its circular path to be constant

• Although speed is constant, velocity is not as along a circular path direction is constantly changing
  
  • Velocity changing= changing acceleration not to change speed of the object but to change direction of the velocity to keep the object on its circular path
    
    • To produce an acceleration there must be a force otherwise the object would move in a straight line (Newton’s first law)

Question 2

Centripetal Acceleration

Answer 2

• Velocity vector is always tangential to the object’s path (regardless of shape of trajectory)
  
  • Δv= v₂ - v₁ point towards the center of the circle in a circular path, because of this so does acceleration as a= Δv/Δt
    
    • Because the acceleration points toward the center of the circle it’s called centripetal acceleration, or a_c
• Centripetal acceleration is what turns the velocity vector to keep the object traveling in a circle
• The magnitude of the centripetal acceleration depends on the object’s speed, v, and the radius, r, of the circular path
  
  • a_c= v²/r

Question 3

Centripetal Force

Answer 3

Using Newton’s second law

F_c= ma_c= mv²/r

• The equation gives the magnitude of the force, as for direction, the directions of F and a are always the same
  
  • Since centripetal acceleration points toward the center of the circular path, so does centripeta force
• Centipetal force is not a new force that makes things move in circles
  
  • Real forces (E.G gravity, friction, tension, normal) provide the centripetal force necessary to maintian circular motion
• All centripetal forces do no work, since the force is directed toward the center and the motion is tangent to the circle (θ= 90⁰)
• If centripetal force is suddently removed, Newton’s first law states that the object would move in a straight line with constant velocity (until another force acts on it)
  
  • The object would move in the direction it was moving when the force was removed (tangent to the circle)

Question 4

Centripetal Force Vs Centrifugal Force

Answer 4

• Centripetal force is the force necessary to maintian circular motion, directed toward the center
  
  • Centrifugal force (does not actually exist) is what an object feels as it moves in a circle (an outward “force”)
    
    • This is not actually a force, but the effect of the object’s inertia (resistance to acceleration)

Question 5

Center of Mass

Answer 5

Point where all of the mass of an object can be considered to be concentrated

• For a homogeneous body, the center of mass is at its geometric center
• For a collection of discrete particles, the center of mass of the system can be determined by selecting some point to be the origin and determine the positions of each particle on the axis, multiply each position value by the mass of the particle at that location and get the sum for all the particles; divide this sum by the total mass and the resulting x value is the center of mass
  
  • x_cm= (m₁x₁ + m₂x₂ + …. m_nx_n)/ (m₁ + m_{2 +} …. m_n)
    
    • The system of particles behave as if all its mass were concentrated at a single location, x_cm
    • If the system consists of objects that are not confined to the same straight line, use the equation to find the x-coordinate of the center of mass and:
      
      • y_cm= (m₁y₁ + m₂y₂ + …. m_ny_n)/(m₁ + m₂ + …. m_n) to find the y coordinate of the center of mass

Question 6

Homogeneous Body

Answer 6

One for which the density is uniform throughout

Question 7

Net External Force

Answer 7

F_net= Ma_cm

• The net (external) force acting on the system causes the center of mass to accelerate according to Newton’s second law
• If the net external force on the system is zero, then the center of mass will not accelerate

Question 8

Rotation and Translation

Answer 8

• All motion is some combination of translation and rotation
• Two points in an object connected by a straight line
  
  • As the object moves, if the line remains parralel to itself while the object is moving, then the object is translating only
  • If the line does not always remain parallel to itself when moving then the object is rotating

Question 9

Translational Dynamics Vs. Rotational Dynamics

Answer 9

• The dynamics of translational motion involve describing the acceleration of an object in terms of its mass (inertia) and the forces that act on it: F_net= ma
• The dynamics of rotational motion involve the angular (rotational) acceleration of an object in terms of its rotational inertia and the torques that act on it

Question 10

Torque

Answer 10

Quantity that measures how effectively a force causes rotation

• Torque isn’t actually a force, but it acts like one
  
  • Torque can be thought of as a turning force, causing objects to rotate either in a clockwise or counterclock wise direction
    
    • Just like a force is a vector quantitiy that produces linear acceleration, a torque is a vector quantity that produces angular acceleration
      
      • A torque can be thought of as positive if it produces a counterclockwise rotation and negative if it produces a clockwise rotation
• To find the net torque acting on an object, simply sum the torques as would for net force
  
  • Counting a counterclockwise torque as positive and a clockwise torque as negative

Question 11

Torque of a Force

Answer 11

Let r be the distance from the pivot (axis of rotation) to the point of application of the force F, and let θ be the angle between vectors r and F

• The magnitude of the torque of F, denoted by τ= rFsinθ
• The angle between two vectors is the angle between then when they start at the same point
  
  • Supplementary angle can be used in the definition of torque as the sine o f an angle and the sine of its supplement are always equal

Question 12

Line of Action

Answer 12

Gives alternative way in determining torque

• The lever arm (or moment arm). l, is the perpendicular distance from the pivot point to the line of action of the force (an extended line along the Force vector)
  
  • The torque of F is defined as the product
    
    • τ= l F
    • τ= l F and τ= rFsinθ as l= rsinθ
• As the lever arm can be called the moment arm, torque can also be called the moment of the force

Question 13

r and F components

Answer 13

• Since l is the component of r that’s perpendicular to F, it is also symbolized by r_⟂​
  
  • ​Definition of torque can also be written as τ= r_⟂F
• Only the component of F that’s perpendicular to r produces torque
  
  • The component of F that’s parallel to r does not produce torque
    
    • τ= rFsinθ= rF_⟂
      
      • Definition of torque can be written as τ= rF_⟂

Question 14

Equilibrium

Answer 14

• An object is said to be in translational equilibrium if the sum of the forces acting on it is zero
  
  • F_net= 0
    
    • a= 0, v is constant
• An object is said to be in rotational equilibrium if the sum of the torques acting on it is zero
  
  • τ_net= 0
    
    • τ_clockwise= τ_{counterclockwise}
• The term equilibrium by itself means both translational and rotational equilibrium are true
  
  • An object in equlibrium may be in motion
    
    • F_net= 0 does not mean that the velocity is zero it means that velocity is constant
    • τ_net= 0 does not mean that the angular velocity is zero, it means that angular velocity is constant

Question 15

Static Equilibrium

Answer 15

An object at rest is said to be in static equilibrium

Question 16

Angular Momentum

Answer 16

• Newton’s second law: F= Δp/Δt= Δ(mv)/Δt
  
  • Multiplying both sides of the equation by r, rF= τ
    
    • τ= Δ(rmv)/Δt
      
      • To form the analog of the law F= Δp/Δt (force equals the rate of change of linear momentum) we say torque equals the rate of change of linear momentum which would rmv
        
        • L= rmv, L denotes angular momentum
          
          • If point mass m does not move in a circular path, angular momentum can be defined relative to a reference point
            
            • If r is the vector from the reference point to the mass, then the angular momentum is: L=rmv_⟂

Question 17

Moment of Inertia

Answer 17

• For a rotating object, the angular momentum equals the sum of the angular momentum of each individual particle
  
  • Written as L= Iw
    
    • I= moment of inertia
      
      • Measure of how difficult it is to start an object rotating, analogous to mass in translational dynamics
        
        • I increases with mass and average radius from the axis of rotation
    • w= angular velocity

Question 18

Conservation of Angular Momentum

Answer 18

Newton’s second law says that F_net= Δp/Δt

If F_net= 0 then p is constant; this is conservation of linear moment

Rotational analog:

τ_net= ΔL/Δt

• If τ_net= 0, then L is constant; this is conservation of angular momentum
  
  • If the torques on a body balance so that the net torque is zero, then the body’s angular momentum won’t change

Question 19

Conservation of Angular Momentum of a Skater

Answer 19

When a figure skater spins, she pulls her arms inwards moving more of her mass closer to the rotation axis and decreasing her rotational inertia, I. Since the external torque on her is negligible, her angluar momentum must be conserved. Since L= Iw, a decrease in I causes an increase in w, and she spins faster

Question 20

Rotational Kinematics

Answer 20

The angular kinematic qualities of θ, w, α, and τ are analogous to the linear kinematics s, v, a, and F respectively

• Average angular velocity:
  
  • ^-w^-= Δθ/Δt
• Average angular acceleration
  
  • ^-α^-= Δw/Δt
• All points on rotating disk undergo same angular displacement during any given time interval, though all points do not have same linear velocity, v

Question 21

Rigid Body

Answer 21

In a rigid body all points along a radial line always have the same angular displacement

Question 22

Angular Acceleration Vs Linear Acceleration

Answer 22

s= rθ; s= arc length

dividing by t

s/t= rθ/t

^-v^-= r^-w^- → v= rw

• The greater the value of r, or v= rw the greater the value of v
  
  • Points on the rotating body farther from the rotation axis move faster than those closer to the rotation axis
  • By deriving the equation above we get
    
    • a= rα
      
      • The acceleration a in this equation is not centripetal acceleration, but tangential acceleration which arises from a change in speed caused by angular acceleration
        
        • Centripetal acceleration does not produce a change in speed
        • Often tangential acceleration is written as a_t to distinguish it from centripetal acceleration a_c

Question 23

Kepler’s First Law

Answer 23

• Every planet moves in an elliptical orbit, with the sun at one focus
  
  • An ellipse has two foci (a pair of special points inside the ellipse such that the sum of every point on the ellipse to the two foci is always the same)

Question 24

Kepler’s Second Law

Answer 24

• As a planet moves in its orbit, a line drawn from the sun to the planet sweeps out equal areas in equal time interals
  
  • The planet moves faster when it is nearer the sun than when it is farther away

Question 25

Kepler’s Third Law

Answer 25

• If T is the period (the time it takes the planet to make one orbit around the sun) and a is the length of the semimajor axis of a planet’s orbit
  
  • Then the ratio T²/a³ is the same for all planets