Answer:
a) 0.25 s
b) 15.71 mm
Step-by-step explanation:
Given:-
- The radius of the coin, r = 10 mm
- The angle swept by the coin, θ = 180°
- The frequency of ration of the coin, f = 2 rev /s
Find:-
How long will it take the coin to roll through the given angle measure at the
given angular velocity?
How far will it travel in that time?
Solution:-
- We will first determine the angular speed ( ω ) of the coin. That is the rate of change of angle swept. Mathematically expressed as:
ω = dθ / dt = 2*π*f
- Separate the variables:
dθ = 2*π*f . dt
- Integrate both sides.
∫dθ = 2*π*f ∫ dt
θ = 2*π*f*t
- The time taken to sweep an angle of ( θ ) is:
t = θ / 2*π*f
Where, θ is in radians. 180° = π radians
- Plug in the values:
t = π / 2*π*( 2 )
t = 1 / 4 = 0.25 s ... Answer
- The tangential speed (v) of any point on the circumference of a coin is. Considering only rolling motion of the coin:
v = r*ω = 2*r*π*f
- The velocity of the any point on the rolling coin circumference would be:
v = 2*(10)*π*(2)
v = 40π mm/s
- Since we are considering the coin as a rigid body and not a point mass. We have to determine the velocity of the center of mass of the coin ( Vcm ).
- Consider a coin as a circle. The point of contact of the between the circle ( coin ) is called the center of instantaneous velocity.
- Then mark two horizontal velocity vectors. One starts at the center of mass of the coin ( pointing right ): Denote this as the velocity of center of mass ( Vcm ).
- Other one starts from top most point lying on the circumference of the circle, this vector should be longer than the one made at center of mass (pointing right ): Denote this as the tangential velocity ( v ).
- Now joint the heads of two vectors ( v and Vcm ) with the center of instantaneous velocity ( contact between coin and surface ). Now make a vertical line that starts the top most point passing through center of mass and ends at the center of instantaneous velocity.
- We will end up with two similar triangles. We will use the law of similar triangle and determine the velocity of center of mass ( Vcm ):
2r / r = v / Vcm
Vcm = v / 2
- Evaluate the velocity of center of mass ( Vcm ):
Vcm = 40π / 2
Vcm = 20π mm/s
- Use the distance-speed-time relationship and determine how far the coin travelled ( s ) in the computed time in part a:
s = Vcm*t
s = 20π*0.25
s = 5π = 15.70796 mm
Answer: The distance travelled by the coin in the time interval of 0.25 seconds is 15.71 mm ( rounded to nearest tenth )
