QuestionFind the time $t$ for a block to slide down an incline with base $a=12$ and angle $\theta=45^{\circ}$ using $t=\sqrt{\frac{2 a}{g \sin \theta \cos \theta}}$ .

Studdy Solution

STEP 1

Assumptions1. The length of the base of the inclined plane, $a$ , is12 feet. The angle of inclination, $\theta$ , is $45^{\circ}$
3. The acceleration due to gravity, $g$ , is approximately32 feet per second per second4. Friction is ignored5. The time $t$ (in seconds) required for a block to slide down an inclined plane is given by the formula $t=\sqrt{\frac{ a}{g \sin \theta \cos \theta}}$

STEP 2

First, we need to convert the angle of inclination from degrees to radians because the trigonometric functions in the formula are usually defined with angles in radians.
$\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}$

STEP 3

Now, plug in the given value for the angle of inclination to convert it to radians.
$\theta_{radians} =45^{\circ} \times \frac{\pi}{180}$

STEP 4

Calculate the angle of inclination in radians.
$\theta_{radians} = \frac{\pi}{4}$

STEP 5

Now, we can substitute the given values and the calculated angle in radians into the formula for the time $t$ .
$t=\sqrt{\frac{2 a}{g \sin \theta_{radians} \cos \theta_{radians}}}$

STEP 6

Plug in the values for $a$ , $g$ , and $\theta_{radians}$ to calculate the time $t$ .
$t=\sqrt{\frac{2 \times12}{32 \times \sin \left(\frac{\pi}{4}\right) \times \cos \left(\frac{\pi}{4}\right)}}$

STEP 7

Calculate the time $t$ .
$t=\sqrt{\frac{24}{32 \times \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}}}$

STEP 8

implify the expression under the square root.
$t=\sqrt{\frac{24}{16}}$

STEP 9

Calculate the time $t$ .
$t=\sqrt{.5}$

STEP 10

Calculate the square root.
$t \approx.2 \, \text{sec}$ So, it takes approximately.2 seconds for the block to slide down the inclined plane.

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QuestionFind the time $t$ for a block to slide down an incline with base $a=12$ and angle $\theta=45^{\circ}$ using $t=\sqrt{\frac{2 a}{g \sin \theta \cos \theta}}$ .

Studdy Solution

STEP 1

STEP 2

First, we need to convert the angle of inclination from degrees to radians because the trigonometric functions in the formula are usually defined with angles in radians.
$\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}$

STEP 3

Now, plug in the given value for the angle of inclination to convert it to radians.
$\theta_{radians} =45^{\circ} \times \frac{\pi}{180}$

STEP 4

Calculate the angle of inclination in radians.
$\theta_{radians} = \frac{\pi}{4}$

STEP 5

Now, we can substitute the given values and the calculated angle in radians into the formula for the time $t$ .
$t=\sqrt{\frac{2 a}{g \sin \theta_{radians} \cos \theta_{radians}}}$

STEP 6

Plug in the values for $a$ , $g$ , and $\theta_{radians}$ to calculate the time $t$ .
$t=\sqrt{\frac{2 \times12}{32 \times \sin \left(\frac{\pi}{4}\right) \times \cos \left(\frac{\pi}{4}\right)}}$

STEP 7

Calculate the time $t$ .
$t=\sqrt{\frac{24}{32 \times \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}}}$

STEP 8

implify the expression under the square root.
$t=\sqrt{\frac{24}{16}}$

STEP 9

Calculate the time $t$ .
$t=\sqrt{.5}$

STEP 10

Calculate the square root.
$t \approx.2 \, \text{sec}$ So, it takes approximately.2 seconds for the block to slide down the inclined plane.

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QuestionFind the time ttt for a block to slide down an incline with base a=12a=12a=12 and angle θ=45∘\theta=45^{\circ}θ=45∘ using t=2agsin⁡θcos⁡θt=\sqrt{\frac{2 a}{g \sin \theta \cos \theta}}t=gsinθcosθ2a​​.

Studdy Solution

STEP 1

STEP 2

First, we need to convert the angle of inclination from degrees to radians because the trigonometric functions in the formula are usually defined with angles in radians.θradians=θdegrees×π180\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}θradians​=θdegrees​×180π​

STEP 3

Now, plug in the given value for the angle of inclination to convert it to radians.θradians=45∘×π180\theta_{radians} =45^{\circ} \times \frac{\pi}{180}θradians​=45∘×180π​

STEP 4

Calculate the angle of inclination in radians.θradians=π4\theta_{radians} = \frac{\pi}{4}θradians​=4π​

STEP 5

Now, we can substitute the given values and the calculated angle in radians into the formula for the time ttt.t=2agsin⁡θradianscos⁡θradianst=\sqrt{\frac{2 a}{g \sin \theta_{radians} \cos \theta_{radians}}}t=gsinθradians​cosθradians​2a​​

STEP 6

Plug in the values for aaa, ggg, and θradians\theta_{radians}θradians​ to calculate the time ttt.t=2×1232×sin⁡(π4)×cos⁡(π4)t=\sqrt{\frac{2 \times12}{32 \times \sin \left(\frac{\pi}{4}\right) \times \cos \left(\frac{\pi}{4}\right)}}t=32×sin(4π​)×cos(4π​)2×12​​

STEP 7

Calculate the time ttt.t=2432×12×12t=\sqrt{\frac{24}{32 \times \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}}}t=32×2​1​×2​1​24​​

STEP 8

implify the expression under the square root.t=2416t=\sqrt{\frac{24}{16}}t=1624​​

STEP 9

Calculate the time ttt.t=.5t=\sqrt{.5}t=.5​

STEP 10

Calculate the square root.t≈.2 sect \approx.2 \, \text{sec}t≈.2secSo, it takes approximately.2 seconds for the block to slide down the inclined plane.

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QuestionFind the time $t$ for a block to slide down an incline with base $a=12$ and angle $\theta=45^{\circ}$ using $t=\sqrt{\frac{2 a}{g \sin \theta \cos \theta}}$ .

First, we need to convert the angle of inclination from degrees to radians because the trigonometric functions in the formula are usually defined with angles in radians.
$\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}$

Now, plug in the given value for the angle of inclination to convert it to radians.
$\theta_{radians} =45^{\circ} \times \frac{\pi}{180}$

Calculate the angle of inclination in radians.
$\theta_{radians} = \frac{\pi}{4}$

Now, we can substitute the given values and the calculated angle in radians into the formula for the time $t$ .
$t=\sqrt{\frac{2 a}{g \sin \theta_{radians} \cos \theta_{radians}}}$

Plug in the values for $a$ , $g$ , and $\theta_{radians}$ to calculate the time $t$ .
$t=\sqrt{\frac{2 \times12}{32 \times \sin \left(\frac{\pi}{4}\right) \times \cos \left(\frac{\pi}{4}\right)}}$

Calculate the time $t$ .
$t=\sqrt{\frac{24}{32 \times \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}}}$

implify the expression under the square root.
$t=\sqrt{\frac{24}{16}}$

Calculate the time $t$ .
$t=\sqrt{.5}$

Calculate the square root.
$t \approx.2 \, \text{sec}$ So, it takes approximately.2 seconds for the block to slide down the inclined plane.