Acceleration with change in direction

[Mavlax]

Homework Statement

Calculate the average acceleration of the tip of the second hand from the moment it hits the 12 to the moment it hits the 3 position. The radius is 5 cm. Its a clock by the way.
I've been at it for over an hour and am beggining to susspect my textbook is wrong.

Homework Equations

a= v2 - v1 / t

The Attempt at a Solution

I got 0.0047 m/s squard.
The textbook says its 4.9 x 10 to the power of negative four [S 45 W]. So 0.00049 m/s squard.

 

[shaurya]

Can you give the detailed explanation?...by the way did you do the conversion right i.e. 5cm=0.05m

 

[The legend]

i solved it and found the text is right ... you haven't done the conversion.
5cm = 0.05m

 

[shaurya]

Yes the answer is correct as per the textbook!

 

[rwisz]

I would first like to commend you for your efforts in attempting to solve this problem. Calculating the acceleration of the second hand on a clock can be tricky, as it involves both a change in direction and a change in speed. It is important to note that the acceleration of an object is not solely dependent on its speed, but also on its change in direction.

In this case, the second hand on a clock has a constant speed of 1 revolution per minute, but it also changes direction every 5 seconds. This means that in order to accurately calculate its acceleration, we must take into account both its change in speed and its change in direction.

Using the equation a= (v2-v1)/t, where v2 is the final velocity, v1 is the initial velocity, and t is the time interval, we can calculate the average acceleration of the second hand from the moment it hits the 12 to the moment it hits the 3 position.

First, let's calculate the change in speed. The second hand travels a distance of 5 cm in 5 seconds, which means its final speed is 5 cm/5s= 1 cm/s. Its initial speed at the 12 position is 0 cm/s. Therefore, the change in speed is 1 cm/s - 0 cm/s = 1 cm/s.

Next, let's calculate the change in direction. The second hand rotates 90 degrees (or 1/4 of a full circle) from the 12 position to the 3 position. This means that its change in direction is 90 degrees.

Now, we can plug in our values into the acceleration equation: a= (1 cm/s - 0 cm/s)/5 s = 0.2 cm/s^2. However, this is only the acceleration in the linear direction. To calculate the total acceleration, we must also take into account the change in direction. In this case, the second hand changes direction by 90 degrees in 5 seconds, which means its angular acceleration is 90 degrees/5s= 18 degrees/s^2.

To convert this angular acceleration to linear acceleration, we must use the formula a= r*alpha, where r is the radius of the circle (5 cm in this case) and alpha is the angular acceleration. Plugging in our values, we get a= 5 cm * (18 degrees/s^2) = 90