Find the angular velocity of a clock's second hand and the linear speed and linear acceleration of a point on the second hand 7.54 cm from the rotation axis.
Well, let's see. The second hand of a clock completes a full revolution in 60 seconds. So, in a minute, it goes around the clock 60 times. That means its angular velocity is a whopping 60 revolutions per minute, or 2π radian per minute.
Now, to convert that to a more practical unit, we divide by 60 (since there are 60 minutes in an hour) to get the angular velocity in radians per second. So, the angular velocity of a clock's second hand is (2π rad/min) / 60 = π/30 rad/s.
As for the linear speed and linear acceleration, we can use a little bit of trigonometry. The distance from the rotation axis to the point on the second hand is given as 7.54 cm. Since we know the angular velocity in radians per second, we can use the formula:
v = ω * r,
where v is the linear speed, ω is the angular velocity, and r is the distance from the rotation axis.
Plugging in the values, we get:
v = (π/30 rad/s) * 7.54 cm = 0.2509 cm/s.
So, the linear speed of a point on the second hand is approximately 0.2509 cm/s.
To find the linear acceleration, we need to use a slightly different formula:
a = α * r,
where a is the linear acceleration and α is the angular acceleration. Since the second hand moves at a constant speed, its angular acceleration is zero. Therefore, the linear acceleration is also zero.
So, in conclusion, the angular velocity of a clock's second hand is π/30 rad/s, the linear speed of a point on the second hand 7.54 cm from the rotation axis is approximately 0.2509 cm/s, and the linear acceleration is zero. Isn't that ticking hilarious?
To find the angular velocity of a clock's second hand, we first need to know the time period it takes for the hand to complete one full revolution. In this case, we can assume it's 60 seconds, as a clock's second hand completes one full revolution in one minute.
The angular velocity (ω) is defined as the change in angular displacement (θ) per unit time (t). In this case, the angular displacement for one full revolution (360 degrees) is 2π radians, and the time period is 60 seconds.
ω = θ / t
ω = 2π radians / 60 seconds
= π / 30 radians per second
Therefore, the angular velocity of the clock's second hand is π/30 radians per second.
To find the linear speed and linear acceleration of a point on the second hand 7.54 cm from the rotation axis, we need to use the formulas relating linear and angular quantities.
The linear speed (v) of a point on a rotating object is given by the product of the radius (r) and the angular velocity (ω).
v = r * ω
v = 7.54 cm * (π/30 radians per second)
≈ 0.7926 cm/s
Therefore, the linear speed of a point on the second hand 7.54 cm from the rotation axis is approximately 0.7926 cm/s.
The linear acceleration (a) of a point on a rotating object is given by the product of the radius (r) and the square of the angular velocity (ω).
a = r * ω^2
a = 7.54 cm * ((π/30 radians per second)^2)
≈ 0.8277 cm/s^2
Therefore, the linear acceleration of a point on the second hand 7.54 cm from the rotation axis is approximately 0.8277 cm/s².