A 4-lb ball b is traveling around in a circle of radius r1 = 3 ft with a speed (vb)1 = 6 ft>s. if the attached cord is pulled
down through the hole with a constant speed vr = 2 ft>s, determine the ball's speed at the instant r2 = 2 ft. how much work has to be done to pull down the cord? neglect friction and the size of the ball
1 answer:
Position #1:
radius, r₁ = 3 ft
Tangential speed, v₁ = 6 ft/s
By definition, the angular speed is
ω₁ = v₁/r₁ = (3 ft/s) / (3 ft) = 1 rad/s
Position #2:
Radius, r₂ = 2 ft
By definition, the moment of inertia in positions 1 and 2 are respectively
I₁ = (4 lb)*(3 ft)² = 36 lb-ft²
I₂ = (4 lb)*(2 ft)² = 16 lb-ft²
Because momentum is conserved,
I₁ω₁ = I₂ω₂
Therefore the angular velocity in position 2 is
ω₂ = (I₁/I₂)ω₁
= (36/16)*1 = 2.25 rad/s
The tangential velocity in position 2 is
v₂ = r₂ω₂ = (2 ft)*(225 rad/s) = 4.5 ft/s
At each position, there is an outward centripetal force.
In position 1, the centripetal force is
F₁ = m*(v²/r₂) = (4)*(6²/3) = 48 lbf
In position 2, the centripetal force is
F₂ = (4)*(4.5²/2) = 40.5 lbf
The radius diminishes at a rate of 2 ft/s.
Therefore the force versus distance curve is as shown below.
The work done is the area under the curve, and it is
W = (1/2)*(48.0+40.5 ft)*(3-2 ft) = 44.25 ft-lb
Answer: 44.25 ft-lb
radius, r₁ = 3 ft
Tangential speed, v₁ = 6 ft/s
By definition, the angular speed is
ω₁ = v₁/r₁ = (3 ft/s) / (3 ft) = 1 rad/s
Position #2:
Radius, r₂ = 2 ft
By definition, the moment of inertia in positions 1 and 2 are respectively
I₁ = (4 lb)*(3 ft)² = 36 lb-ft²
I₂ = (4 lb)*(2 ft)² = 16 lb-ft²
Because momentum is conserved,
I₁ω₁ = I₂ω₂
Therefore the angular velocity in position 2 is
ω₂ = (I₁/I₂)ω₁
= (36/16)*1 = 2.25 rad/s
The tangential velocity in position 2 is
v₂ = r₂ω₂ = (2 ft)*(225 rad/s) = 4.5 ft/s
At each position, there is an outward centripetal force.
In position 1, the centripetal force is
F₁ = m*(v²/r₂) = (4)*(6²/3) = 48 lbf
In position 2, the centripetal force is
F₂ = (4)*(4.5²/2) = 40.5 lbf
The radius diminishes at a rate of 2 ft/s.
Therefore the force versus distance curve is as shown below.
The work done is the area under the curve, and it is
W = (1/2)*(48.0+40.5 ft)*(3-2 ft) = 44.25 ft-lb
Answer: 44.25 ft-lb
You might be interested in
Answer:
The rotational inertia of the pendulum around its pivot point is .
Explanation:
The angular frequency of a physical pendulum is measured by the following expression:
Where:
- Angular frequency, measured in radians per second.
- Mass of the physical pendulum, measured in kilograms.
- Gravitational constant, measured in meters per square second.
- Straight line distance between the center of mass and the pivot point of the pendulum, measured in meters.
- Moment of inertia with respect to pivot point, measured in
.
In addition, frequency and angular frequency are both related by the following formula:
Where:
- Frequency, measured in hertz.
If , then angular frequency of the physical pendulum is:
From the formula for the physical pendulum's angular frequency, the moment of inertia is therefore cleared:
Given that ,
,
and
, the moment of inertia associated with the physical pendulum is:
The rotational inertia of the pendulum around its pivot point is .
As per momentum conservation theory
If net force on a system is zero then initial momentum of the system will be equal to final momentum of that system.
Since initially coconut is at rest so initial momentum is zero
Here it break into three pieces such that two parts of same mass while third part is of double mass
now we can say
here we know that two parts fly off with same speed v0 along south and west directions
now we will have
so here magnitude of the speed of the third part will be
and the direction of the third part will be along North East direction