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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.3, Problem 36E

(a)

To determine

**To find:** The velocity and acceleration at time *t*.

Expert Solution

The velocity of motion at time *t* is

The acceleration of motion at time *t* is

**Given:**

The equation of motion is *t* is in seconds and *x* is in centimeters.

**Derivative rule:**

(1) Sum Rule:

(2) Constant Multiple Rule:

**Recall:**

If *t* is in seconds, then the velocity of the particle is

If *t* is in seconds, then the acceleration of the particle is

**Calculation:**

Obtain the velocity at time *t.*

Apply the sum rule (1),

Apply the constant multiple rule (2),

Thus, the velocity of *t* is

Obtain the acceleration at time *t.*

Apply the sum rule (1)

Apply the constant multiple rule (2)

Therefore, the acceleration of *t* is

(b)

To determine

**To sketch:** The velocity and acceleration functions.

Expert Solution

From part (a), the velocity and acceleration functions are

Use online graphing calculator and draw graph the velocity and acceleration functions as shown below in Figure 1.

From Figure1, it is observed that the maximum position of the mass travel is approximately

**(**c**)**

To determine

**To find:** When the mass passes through the equilibrium position for the first time.

Expert Solution

The mass passes through the equilibrium position for the first time is

**Definition used:**

The mass is in equilibrium when its acceleration is zero.

**Calculation:**

The equation of motion is

From (a), the acceleration of *t* is

Use the definition stated above and solve for

Multiply the equation by

Therefore, the mass of equilibrium at *n* is an integer.

Compute *t* when

Hence, the mass passes through the equilibrium for the first time is

(d)

To determine

**To find:** The distance from the equilibrium position to the final position.

Expert Solution

The mass travels about 3.6056 cm from its equilibrium position.

From Figure1, the maximum position is

From part (a),

Set *t*.

Multiply the equation by

Obtain

Thus, the mass travels about 3.6056 cm from its equilibrium position.

(e)

To determine

**To find:** When is the speed greatest?

Expert Solution

The speed is maximum at

Note that, the velocity is zero when the mass reaches its maximum or minimum and the speed will be greatest when the mass passess through the equilibrium position.

That is,

Multiply the equation by

For each additional rotation of *n* is an integer.