The hour hand on a certain clock is 8.3 cm long.
Find the tangential speed of the tip of this hand.
v{t}= ???? {mm/s}
.0083*(2pi^2)/2
oops that should be (.0083*2pi)^2/2
NOT .0083*(2pi^2)/2
To find the tangential speed of the tip of the hour hand, we need to determine the distance traveled by the tip in a given time.
The formula for tangential speed is:
v_t = r * ω
where v_t is the tangential speed, r is the radius of the circular path, and ω is the angular velocity.
In this case, the radius of the circular path is the length of the hour hand, which is given as 8.3 cm.
To find the angular velocity, we need to know the rate at which the hour hand moves. Since we don't have that information, we'll assume that the hour hand completes one revolution every 12 hours.
So, the angular velocity ω can be calculated as follows:
ω = (2π radians) / (12 hours)
Now we can substitute the values into the formula to find the tangential speed:
v_t = (8.3 cm) * [(2π radians) / (12 hours)]
To convert the mm/s, we need to convert cm to mm and hours to seconds.
1 cm = 10 mm and 1 hour = 3600 seconds.
v_t = (8.3 cm) * [(2π radians) / (12 hours)] * (10 mm/cm) * (3600 seconds/hour)
Simplifying this expression gives:
v_t ≈ 4,598 mm/s
Therefore, the tangential speed of the tip of the hour hand is approximately 4,598 mm/s.
To find the tangential speed of the tip of the hour hand, we can use the formula:
vₜ = r × ω
Where:
- vₜ is the tangential speed
- r is the radius (length) of the hour hand
- ω is the angular speed in radians per second
In this case, we are given that the length of the hour hand (r) is 8.3 cm. However, to express the answer in mm/s, it is better to convert this length to millimeters:
r = 8.3 cm × 10 mm/cm
r = 83 mm
Now we need to find the value of the angular speed (ω). The angular speed of the hour hand is typically given in degrees per hour or radians per hour. However, to express the answer in mm/s, we need to convert this unit of time to seconds.
\(1\) hour = \(60\) minutes × \(60\) seconds
\(1\) hour = \(3600\) seconds
For a clock, the hour hand completes a full rotation (360°) in 12 hours. To find the angular speed in radians per second, we can use the formula:
ω = (θ × 2π) / t
Where:
- θ is the angle in radians (360° in this case)
- t is the time in seconds (12 hours in this case)
θ = 360° × (π/180°)
θ ≈ 6.28 radians
t = 12 hours × 3600 seconds/hour
t = 43200 seconds
Now we can calculate the angular speed:
ω = (6.28 radians × 2π) / 43200 seconds
ω ≈ 0.0002909 radians/second
Finally, we can substitute the values into the formula for tangential speed:
vₜ = 83 mm × 0.0002909 radians/second
vₜ ≈ 0.0241 mm/s
Therefore, the tangential speed of the tip of the hour hand is approximately 0.0241 mm/s.