All AP Physics 1 Resources
Example Questions
Example Question #11 : Circular And Rotational Motion
A spinning disk is rotating at a rate of in the positive counterclockwise direction. If the disk is speeding up at a rate of , find the disk's angular velocity in after four seconds.
None of these
The angular velocity is given by:
Example Question #11 : Circular And Rotational Motion
A model train completes a circle of radius in . Determine the angular frequency in .
None of these
One circle is equal to , thus
Example Question #11 : Circular And Rotational Motion
Two cars are racing side by side on a circular race track. Which has the greater angular velocity?
They are the same
Impossible to determine
The outside car
The inside car
They are the same
If the cars are racing side by side on a circular track, then they have the same angular velocity, because they complete their circles in the same amount of time.
Example Question #11 : Angular Velocity And Acceleration
Matthew is swinging a bucket () in a vertical circle via a rope () as shown in the figure. At the bottom of the circular path, the rope's tension is . What is the bucket's speed at this time?
This is a centripetal force problem. It's important to know that when you draw your free body diagram (FBD), centripetal force () isn't drawn, similar to how net forces () aren't drawn. is actually the centripetal or circular form of. So when we talk about circular motion such as this, we can set the two equal to each other. Why is this? Well we are trying to translate this object's circular motion into it's linear velocity (our answer).
We can now set, or equal to the combination of vertical forces in our FBD
But like we said earlier, we need to connect our linear forces to our centripetal forces. Remember this equation?
Now let's put it all together
Time to pull in some algebra now. Remember we want the speed of the bucket, or the velocity of the bucket at the lowest point of the circle, so solve for :
If you plut in everything and solve (remember to use PEMDAS), your answer should be
Example Question #18 : Circular And Rotational Motion
A yo-yo professional is doing a bunch of tricks to show off to his new girlfriend. He manages to do an super-mega around-the-world which entails swinging the yo-yo (, string = = ) around the path shown below 4 times in .
What is the centripetal acceleration of the yo-yo?
Centripetal acceleration is as follows:
All we need to do is solve and plug in for
= distance traveled and t is the time it took to travel that distance
We know that the yo-yo made 4 revolutions in .
One revolution is the circumference of the yo-yo path:
So we can either solve for in terms of how far the yo-yo went in one second (I), or how long it took to make one revolution (II); I'll show both, either one is correct if you wish to plug in
or
REMEMBER: Both of these will give you the same answer (check it out if you're good at algebra). Physics is all about comfort in manipulation so choose the way that suits you!
Now plug this into our equation. Lucky for you, the radius was
Note: I put in the units so you can see how they cancel out algebraically
Example Question #1 : Angular Momentum
In an isolated system, the moment of inertia of a rotating object is halved. What happens to the angular velocity of the object?
It is doubled.
It is quartered.
It is halved.
It remains the same.
It is quadrupled.
It is doubled.
In an isolated system, there is no net torque. If there is no net torque on the system, then the total angular momentum of the system remains the same. The angular momentum of a rotating object is equal to the moment of inertia of the object multiplied by the object's angular velocity.
is the symbol for angular momentum, is the moment of inertia, and is the angular velocity.
Therefore, if the moment of inertia, , is halved, then for the angular momentum, , to remain constant, the angular velocity, , must be doubled. This is because , which is the multiplicative identity. Anything multiplied by one remains the same. So, the final angular velocity would be twice as large as it was originally.
Example Question #1 : Angular Momentum
I start pushing a merry-go-round with a torque of 10 Newton-meters. It has a moment of inertia of . What is its rotational speed after 3 seconds assuming it starts at rest?
The angular moment of the merry-go-round after 3 seconds is simply
Angular momentum is also given by
Plugging in 30 for gives us
Example Question #1 : Angular Momentum
A satellite is rotating once per minute. It has an moment of inertia of . Erin, an astronaut, extends the satellite's solar panels, increasing its moment of inertia to . How quickly is the satellite now rotating?
1 rotation every 3 minutes
1 rotation per minute
1 rotation every 9 minutes
3 rotations per minute
9 rotations per minute
1 rotation every 3 minutes
The formula for angular momentum is
where
= angular momentum
= moment of inertia
= angular speed
Angular speed is defined as
The initial period of the satellite is 1 minute, so:
Plugging this in, we can solve for the initial angular momentum:
After Erin extends the the solar panels, momentum is the same (conservation of momentum), but the moment of inertia is now . Thus,
Therefore,
Plugging this back into the definition, we get
Therefore,
Thus, the satellite now rotates once every 3 minutes.
Example Question #71 : Circular, Rotational, And Harmonic Motion
A blob of clay of mass is dropped on top of a rotating disk's edge of mass spinning at a speed of . What will the resulting rotational speed be?
Angular momentum is always conserved.
The equation relating angular momentum, moment of inertia and angular velocity is:
Moment of inertia can also be written as follows:
Plug in given values to find and
Set the initial angular momentum equal to the final angular momentum and solve.
Example Question #1 : Angular Momentum
A rectangular rod on a horizontal table top is shown in the left side of the diagram. The rod is hinged at one end, as shown. It rotates in such a way that it hits a stationary ball as shown in the diagram's right side. The rod rotates without friction at a rate of until it contacts the ball. As a result of the collision, the rod comes to rest and the ball moves to the right. The mass of the rod is 0.2kg. The mass of the ball is 0.067kg. The length of the rod is 0.15m. The moment of inertia of a rod rotated about the end is: . What is the ball's speed as a result of the collision?
This is a conservation of angular momentum problem, so we set the angular momentum of the rod to the equivalent angular momentum of the ball
Note that the here is length, not angular momentum. Be careful not to cancel the since it refers to the rod on the left and the ball on the right. Finally, the on the right is the effective radius of the ball at the moment of impact, so it's simply the length of the rod.
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