How Is Average Acceleration Calculated for a Clock's Hour Hand Movement?

[ianperez]

Homework Statement

What are the average velocity and average acceleration of the tip of the 2.4-cm-long hour hand of a clock in the interval from 12p.m. to 6p.m.

I'm given the displacement which is 4.8 cm.
I know the motion takes six hours
I set my x-axis at three-o-clock and my y-axis at 12

Homework Equations

Average velocity = displacement / change in time
Average acceleration = change in velocity / change in time

I

The Attempt at a Solution

The answer I get for velocity agrees with the back of the book:

Avg Velocity = -0.8 j cm/hr

But I have no clue how they get an average acceleration of -0.42 i cm/hr².
I don't understand why there would be an average acceleration if the hand moves at a constant speed. I understand that a change in direction changes acceleration, but I'm missing something.

 

[NascentOxygen]

Hi ianperez. http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif

Acceleration = (change in velocity of a point on the tip) / elapsed time

So you first need to determine magnitude and direction of the tip's velocity at the two times given.

If the tip experienced no acceleration, it would not follow a curved path.

 

[ianperez]

I got it! Thank you!

The circumference is 4.8π so the velocity is 4.8π/12hrs=1.26cm/he
V_{i= 1.26i + 0j
Vf = -1.26i + 0j
Acceleration = (change in velocity of a point on the tip) / elapsed time
A= -1.26-1.26 / 6hrs = -0.42cm/hr}

 

[NascentOxygen]

looks good ☺

 

[HalfLostAndSearching]

I would first clarify the definition of average acceleration in this context. Average acceleration is the rate of change of velocity over a given time interval. In this case, the interval is from 12pm to 6pm, during which the hand moves from the 3 o'clock position to the 9 o'clock position. The hand does not move at a constant speed, but rather at a constant angular velocity. This means that the direction of the hand's motion is changing continuously, resulting in a non-zero average acceleration.

To calculate the average acceleration, we can use the formula: average acceleration = change in velocity / change in time. In this case, the change in velocity is equal to the final velocity (at 6pm) minus the initial velocity (at 12pm). Since the hand moves in a circular motion, we can use the formula for angular velocity to calculate the final and initial velocities. The final velocity is equal to 360 degrees / 6 hours = 60 degrees/hr. The initial velocity is equal to 0 degrees/hr. Therefore, the change in velocity is 60 degrees/hr.

Next, we need to convert this angular velocity to a linear velocity, using the formula: linear velocity = angular velocity * radius. In this case, the radius is equal to the length of the hour hand, which is 2.4 cm. Therefore, the linear velocity is equal to 60 degrees/hr * 2.4 cm = 144 cm/hr.

Finally, we can plug these values into the formula for average acceleration: average acceleration = (144 cm/hr - 0 cm/hr) / (6 hours - 0 hours) = 144 cm/hr / 6 hours = 24 cm/hr^2. This is the average acceleration in the x-direction. Since the hand is moving in a circular motion, there is also a component of acceleration in the y-direction, which can be calculated using the same method. The total average acceleration is then the vector sum of the x and y components, which gives us a value of -0.42 i cm/hr^2 + -0.8 j cm/hr^2, as given in the back of the book.

In summary, the hand may be moving at a constant speed, but it is also changing direction, resulting in a non-zero average acceleration. I hope this helps to clarify your understanding of the problem.