Question

The radius of a body moving in a circle with constant angular velocity is given by $r=4t^2$, with respect to time. What is the magnitude of the tangential velocity at $t=4$ if the angular velocity is 7 rad/s?

• A. 50
• B. 112
• C. 120
• D. 65

Answer 1

Answer: (B)

112

Answer 2

The problem asks for the magnitude of the tangential velocity of a body moving in a circle.

1. Identify the given information:

• Radius as a function of time: $r = 4t^2$
• Constant angular velocity: $\omega = 7$ rad/s
• Specific time: $t = 4$ s

2. Recall the formula for tangential velocity: The magnitude of tangential velocity ($v_t$) is related to the radius ($r$) and angular velocity ($\omega$) by the equation: $v_t = r\omega$

3. Calculate the radius at the specified time ($t=4$ s): Substitute $t=4$ into the given radius equation: $r(t=4) = 4 \times (4)^2$ $r(t=4) = 4 \times 16$ $r(t=4) = 64 \text{ units}$

4. Calculate the tangential velocity using the calculated radius and given angular velocity: $v_t = r(t=4) \times \omega$ $v_t = 64 \times 7$ $v_t = 448 \text{ units/s}$

5. Compare the result with the given options: The calculated tangential velocity is 448 units/s. However, this value is not among the provided options (50, 112, 120, 65). This indicates a potential typo in the question's given radius formula.

6. Consider a likely typo that leads to one of the options: Let's examine if a common or simple typo could lead to one of the options. If the radius equation was intended to be $r=4t$ instead of $r=4t^2$: At $t=4$ s, $r = 4 \times 4 = 16$ units. Then, $v_t = 16 \times 7 = 112$ units/s. This matches one of the options.

Alternatively, if the radius equation was intended to be $r=t^2$ instead of $r=4t^2$: At $t=4$ s, $r = (4)^2 = 16$ units. Then, $v_t = 16 \times 7 = 112$ units/s. This also matches one of the options.

Since two plausible typos lead to the same option (112), it is highly probable that 112 is the intended answer. We proceed with the assumption that the radius at $t=4$ was intended to be 16 units.

Final Calculation (assuming intended radius at t=4 is 16 units): Given $r = 16$ units (at $t=4$) and $\omega = 7$ rad/s. $v_t = r\omega$ $v_t = 16 \times 7$ $v_t = 112 \text{ units/s}$

The final answer is 112

Explanation of the solution:

The tangential velocity ($v_t$) is given by the product of the radius ($r$) and the angular velocity ($\omega$), i.e., $v_t = r\omega$. Given $r = 4t^2$ and $\omega = 7$ rad/s. At $t=4$ s, the radius would be $r = 4 \times (4)^2 = 4 \times 16 = 64$ units. Using this, $v_t = 64 \times 7 = 448$ units/s. Since 448 is not an option, a typo in the question is highly probable. If the radius at $t=4$ was intended to be 16 units (which would result from $r=4t$ or $r=t^2$), then $v_t = 16 \times 7 = 112$ units/s, which is an option.