Question

The angular acceleration of a wheel is given in rad/s2 by 32 t 3 - 11t 4, where t is in seconds. if the wheel starts from rest at t = 0.00 s, when is the next time the wheel is at rest?

Answer 1

Answer:

$t=3.64s$

Step by step explanation

Step 1

In this step we use the definition of acceleration to determine the expression to integrate. Note that the acceleration is the derivative of velocity with respect to time.

$a=\frac{dv}{dt} \\\implies adt=dv\\\implies dv=(32t^3-11t^4)dt$

Step 2

Perform  integration on the expression from step 1. This calculation is performed as shown below.

$v=\int(32t^3-11t^4)dt\\v=\frac{32}{4}t^4- \frac{11}{5}t^5 +c\\\8t^4- \frac{11}{5}t^5 +c$

Step 3

In this step we use the condition that at $t=0$, the velocity $v=0$ to find the exact form of the velocity function. We substitute  this point into the velocity function we found in step 2.

$0=8(0)^4-\frac{11}{5} (0)^5+c\\\implies c=0$

Step 4

We now  use the function in step 3  to find out when the velocity is zero again.

$v=0=8t^5-\frac{11}{5}t^4\\\implies 0=8t^4-\frac{11}{5} t^5\\\implies t^4(8-\frac{11}{5} t)\\\implies t=0, 8-\frac{11}{5}t=0\\\\\implies t=0,t=\frac{40}{11} =3.64$

The velocity is zero again when $t=3.64$