The expression for rotational kinetic energy is exactly analogous to translational kinetic energy, with
being analogous to
and
to
. Rotational kinetic energy has important effects. Flywheels, for example, can be used to store large amounts of rotational kinetic energy in a vehicle, as seen in
[link] .
Calculating the work and energy for spinning a grindstone
Consider a person who spins a large grindstone by placing her hand on its edge and exerting a force through part of a revolution as shown in
[link] . In this example, we verify that the work done by the torque she exerts equals the change in rotational energy. (a) How much work is done if she exerts a force of 200 N through a rotation of
? The force is kept perpendicular to the grindstone’s 0.320-m radius at the point of application, and the effects of friction are negligible. (b) What is the final angular velocity if the grindstone has a mass of 85.0 kg? (c) What is the final rotational kinetic energy? (It should equal the work.)
Strategy
To find the work, we can use the equation
. We have enough information to calculate the torque and are given the rotation angle. In the second part, we can find the final angular velocity using one of the kinematic relationships. In the last part, we can calculate the rotational kinetic energy from its expression in
.
Solution for (a)
The net work is expressed in the equation
where net
is the applied force multiplied by the radius
because there is no retarding friction, and the force is perpendicular to
. The angle
is given. Substituting the given values in the equation above yields
Noting that
,
Solution for (b)
To find
from the given information requires more than one step. We start with the kinematic relationship in the equation
Note that
because we start from rest. Taking the square root of the resulting equation gives
Now we need to find
. One possibility is
where the torque is
The formula for the moment of inertia for a disk is found in
[link] :
Substituting the values of torque and moment of inertia into the expression for
, we obtain
Now, substitute this value and the given value for
into the above expression for
:
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits