Deriving newton’s second law for rotation in vector form
As before, when we found the angular acceleration, we may also find the torque vector. The second law
tells us the relationship between net force and how to change the translational motion of an object. We have a vector rotational equivalent of this equation, which can be found by using
[link] and
[link] .
[link] relates the angular acceleration to the position and tangential acceleration vectors:
We form the cross product of this equation with
and use a cross product identity (note that
):
We now form the cross product of Newton’s second law with the position vector
Identifying the first term on the left as the sum of the torques, and
as the moment of inertia, we arrive at Newton’s second law of rotation in vector form:
This equation is exactly
[link] but with the torque and angular acceleration as vectors. An important point is that the torque vector is in the same direction as the angular acceleration.
Applying the rotational dynamics equation
Before we apply the rotational dynamics equation to some everyday situations, let’s review a general problem-solving strategy for use with this category of problems.
Problem-solving strategy: rotational dynamics
Examine the situation to determine that torque and mass are involved in the rotation. Draw a careful sketch of the situation.
Determine the system of interest.
Draw a free-body diagram. That is, draw and label all external forces acting on the system of interest.
Identify the pivot point. If the object is in equilibrium, it must be in equilibrium for all possible pivot points––chose the one that simplifies your work the most.
Apply
, the rotational equivalent of Newton’s second law, to solve the problem. Care must be taken to use the correct moment of inertia and to consider the torque about the point of rotation.
As always, check the solution to see if it is reasonable.
Calculating the effect of mass distribution on a merry-go-round
Consider the father pushing a playground merry-go-round in
[link] . He exerts a force of 250 N at the edge of the 200.0-kg merry-go-round, which has a 1.50-m radius. Calculate the angular acceleration produced (a) when no one is on the merry-go-round and (b) when an 18.0-kg child sits 1.25 m away from the center. Consider the merry-go-round itself to be a uniform disk with negligible friction.
Strategy
The net torque is given directly by the expression
, To solve for
, we must first calculate the net torque
(which is the same in both cases) and moment of inertia
I (which is greater in the second case).
Solution
The moment of inertia of a solid disk about this axis is given in
[link] to be
We have
and
, so
To find the net torque, we note that the applied force is perpendicular to the radius and friction is negligible, so that
Now, after we substitute the known values, we find the angular acceleration to be
We expect the angular acceleration for the system to be less in this part because the moment of inertia is greater when the child is on the merry-go-round. To find the total moment of inertia
I , we first find the child’s moment of inertia
by approximating the child as a point mass at a distance of 1.25 m from the axis. Then
The total moment of inertia is the sum of the moments of inertia of the merry-go-round and the child (about the same axis):
Substituting known values into the equation for
α gives
Questions & Answers
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product