Find the angular velocity of a clock's second hand and the linear speed and linear acceleration of a point on the second hand 1.96 cm from the rotation axis.
To find the angular velocity of a clock's second hand, we need to know the time it takes for the second hand to make a complete revolution. In a standard clock, this time is 60 seconds.
Angular velocity (ω) represents the rate at which an object rotates and is measured in radians per second (rad/s). To calculate the angular velocity of the second hand, we'll use the formula:
ω = 2π / T
where ω is the angular velocity and T is the time taken for one complete revolution.
In this case, T = 60 seconds, so the formula becomes:
ω = 2π / 60
Simplifying, we get:
ω = π / 30 rad/s
Now, let's move on to finding the linear speed and linear acceleration of a point on the second hand located 1.96 cm from the rotation axis.
Linear speed (v) is the distance traveled per unit of time along a circular path and is given by the formula:
v = rω
where v is the linear speed, r is the radius or distance from the rotation axis, and ω is the angular velocity.
In this case, r = 1.96 cm (convert to meters for consistency) and ω = π / 30 rad/s. Substituting the values into the formula:
v = 1.96 * (π / 30)
Calculating, we find:
v ≈ 0.206 m/s
To find the linear acceleration, we use the equation:
a = rω²
where a is the linear acceleration, r is the radius, and ω is the angular velocity.
Substituting the known values:
a = 1.96 * (π / 30)²
Calculate:
a ≈ 0.0218 m/s²
So, the angular velocity of the clock's second hand is approximately π / 30 rad/s, the linear speed of a point 1.96 cm from the rotation axis is approximately 0.206 m/s, and the linear acceleration is approximately 0.0218 m/s².
To find the angular velocity of the clock's second hand, we can use the formula:
Angular velocity (ω) = 2π ÷ Time period (T)
The time period for the clock's second hand is 60 seconds since it completes one full rotation every minute. Therefore, T = 60 seconds.
Plugging in the values, we have:
ω = 2π ÷ 60 = π ÷ 30 rad/s
The angular velocity of the clock's second hand is π ÷ 30 rad/s.
To find the linear speed (v) of a point on the second hand 1.96 cm from the rotation axis, we can use the formula:
Linear speed (v) = Angular velocity (ω) × Radius (r)
Given that r = 1.96 cm, and the angular velocity is π ÷ 30 rad/s, we can calculate the linear speed:
v = (π ÷ 30) × 1.96 = 0.0652 π cm/s
Therefore, the linear speed of a point on the second hand 1.96 cm from the rotation axis is approximately 0.0652 π cm/s.
Finally, to find the linear acceleration (a) of the point on the second hand, we can use the formula:
Linear acceleration (a) = (Angular acceleration (α)) × Radius (r)
Since the angular acceleration of the second hand is assumed to be zero (as it moves with constant speed), the linear acceleration will also be zero.