Question

Physics speed vs time graph?

The figure gives angular speed versus time for a thin rod that rotates around one end. (a) What is the magnitude of the rod's angular acceleration? (b) At t = 4.0 s, the rod has a rotational kinetic energy of 5.08 J. What is its kinetic energy at t = 0?

I just need answer for b
the answer for a is 1.5

Answer 1

See the answer below....

Explanation:

• The graph cuts two point `color(red)((2,1);(4,4)`. So, the equation of the straight line of the graph is

`(y-4)/(x-4)=(4-1)/(4-2)`

`=>3x-2y=4`

`y=underbrace(color(red)(3/2)) x-2`

The gradient that is the value of angular acceleration is `color(red)(1.5)" rad/s".`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let, at `t=0`, the angular velocity is `omega_1` and at `t=4`, the angular velocity is `omega_2`
The formula to determine the rotational kinetic energy is

`color(red)("E"_k=1/2 m omega^2r^2" r" "="radius"`

I can now write,

`1/2 m omega_2^2r^2=5.08`

`=>1/2 m xx4^2xxr^2=5.08" ""as "omega_2=4`

`=>mr^2=127/200`

In the equation, I determined, taking `x(t)=0`, we get `y(omega_1)=-2`.

The kinetic energy at `t=0` is `=>"E"_k=color(green)(1/2xxmr^2 xx omega_1^2)=1/2xx127/200xx(-2)^2=color(red)(2.54" J"`

Hope it helps...
Thank you..