An object with a mass of $2 k g$ $2 kg$ is traveling in a circular path of a radius of $4 m$ $4 m$ . If the object's angular velocity changes from $4 H z$ $4 Hz$ to $5 H z$ $5 Hz$ in $3 s$ $3 s$ , what torque was applied to the object?

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An object with a mass of $2 k g$ $2 kg$ is traveling in a circular path of a radius of $4 m$ $4 m$ . If the object's angular velocity changes from $4 H z$ $4 Hz$ to $5 H z$ $5 Hz$ in $3 s$ $3 s$ , what torque was applied to the object?

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Answer 1 · 2017-09-14T01:37:34

We will need to know the angular acceleration that existed during those 3 s. If this was a linear motion problem we could use the suvat formula
$v = u + a \cdot t$ $v = u + a*t$
The equivalent formula in angular motion is
$ω_{f} = ω_{i} + α \cdot t$ $omega_f = omega_i + alpha*t$

Using that last formula,
$5 \cdot \frac{2 \cdot π radians}{s} = 4 \cdot \frac{2 \cdot π radians}{s} + α \cdot 3 s$ $5*(2*pi " radians")/s = 4*(2*pi " radians")/s + alpha*3 s$

Solving for $α$ $alpha$ ,
$α = \frac{\frac{2 \cdot π radians}{s}}{3 s} = 2.09 \frac{radians}{s^{2}}$ $alpha = ((2*pi " radians")/s)/(3 s) = 2.09 " radians"/s^2$

Next we need the angular motion equivalent of F=m*a (Newton's 2nd Law). The equivalent is

$τ = I \cdot α$ $tau = I*alpha$

I will assume the object can be considered a point mass. This requires that the dimensions of the object be much smaller than the 4 m radius. With that assumption, the rotational inertia, $I$ $I$ , of the rotating mass is
$I = m \cdot r^{2} = 2 k g \cdot {(4 m)}^{2} = 32 k g \cdot m^{2}$ $I = m*r^2 = 2 kg*(4 m)^2 = 32 kg*m^2$

So now we can calculate the torque, $τ$ $tau$ .

$τ = 32 k g \cdot m^{2} \cdot 2.09 \frac{radians}{s^{2}} = 67 k g \cdot \frac{m^{2}}{s^{2}}$ $tau = 32 kg*m^2 * 2.09 " radians"/s^2 = 67 kg*m^2/s^2$
$τ = 67 N \cdot m$ $tau = 67 N*m$

You might ask how the combination of units $k g \cdot \frac{m^{2}}{s^{2}}$ $kg*m^2/s^2$ simplified into $N \cdot m$ $N*m$ . Note that you can group $k g \cdot \frac{m}{s^{2}}$ $kg*m/s^2$ apart from the other m. And $k g \cdot \frac{m}{s^{2}}$ $kg*m/s^2$ is the definition of the Newton.

Torque is the product of force and the length of the lever arm. So, we have our answer.

I hope this helps,
Steve

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An object with a mass of $2 k g$ $2 kg$ is traveling in a circular path of a radius of $4 m$ $4 m$ . If the object's angular velocity changes from $4 H z$ $4 Hz$ to $5 H z$ $5 Hz$ in $3 s$ $3 s$ , what torque was applied to the object?

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An object with a mass of $2 k g$ $2 kg$ is traveling in a circular path of a radius of $4 m$ $4 m$ . If the object's angular velocity changes from $4 H z$ $4 Hz$ to $5 H z$ $5 Hz$ in $3 s$ $3 s$ , what torque was applied to the object?