To determine which of the given statements are correct, we analyze the motion of the body under the given time-dependent force.
1. Acceleration, Velocity, and Displacement Given:
Mass of the body, m = 1.0 kg m = 1.0\text{ kg} m = 1.0 kg
Force acting on the body, F ⃗ = α t i ^ + β j ^ = ( t i ^ + j ^ ) N \vec{F} = \alpha t\,\hat{i} + \beta\,\hat{j} = (t\,\hat{i} + \hat{j})\text{ N} F = α t i ^ + β j ^ = ( t i ^ + j ^ ) N α = 1.0 N s − 1 \alpha = 1.0\text{ N s}^{-1} α = 1.0 N s − 1 β = 1.0 N \beta = 1.0\text{ N} β = 1.0 N
Initial conditions at t = 0 t = 0 t = 0 r ⃗ ( 0 ) = 0 ⃗ \vec{r}(0) = \vec{0} r ( 0 ) = 0 v ⃗ ( 0 ) = 0 ⃗ \vec{v}(0) = \vec{0} v ( 0 ) = 0
From Newton's second law, the acceleration a ⃗ ( t ) \vec{a}(t) a ( t ) a ⃗ ( t ) = F ⃗ ( t ) m = t i ^ + j ^ 1.0 = t i ^ + j ^ m s − 2 \vec{a}(t) = \frac{\vec{F}(t)}{m} = \frac{t\,\hat{i} + \hat{j}}{1.0} = t\,\hat{i} + \hat{j}\text{ m s}^{-2} a ( t ) = m F ( t ) = 1.0 t i ^ + j ^ = t i ^ + j ^ m s − 2
Integrating the acceleration with respect to time to find the velocity v ⃗ ( t ) \vec{v}(t) v ( t ) v ⃗ ( t ) = ∫ 0 t a ⃗ ( t ′ ) d t ′ = ∫ 0 t ( t ′ i ^ + j ^ ) d t ′ = 1 2 t 2 i ^ + t j ^ \vec{v}(t) = \int_0^t \vec{a}(t')\,dt' = \int_0^t (t'\,\hat{i} + \hat{j})\,dt' = \frac{1}{2}t^2\,\hat{i} + t\,\hat{j} v ( t ) = ∫ 0 t a ( t ′ ) d t ′ = ∫ 0 t ( t ′ i ^ + j ^ ) d t ′ = 2 1 t 2 i ^ + t j ^
At t = 1.0 s t = 1.0\text{ s} t = 1.0 s v ⃗ ( 1.0 ) = 1 2 ( 1.0 ) 2 i ^ + ( 1.0 ) j ^ = 1 2 i ^ + j ^ = 1 2 ( i ^ + 2 j ^ ) m s − 1 \vec{v}(1.0) = \frac{1}{2}(1.0)^2\,\hat{i} + (1.0)\,\hat{j} = \frac{1}{2}\hat{i} + \hat{j} = \frac{1}{2}(\hat{i} + 2\hat{j})\text{ m s}^{-1} v ( 1.0 ) = 2 1 ( 1.0 ) 2 i ^ + ( 1.0 ) j ^ = 2 1 i ^ + j ^ = 2 1 ( i ^ + 2 j ^ ) m s − 1 Option C is correct .
Now, integrating the velocity to find the position vector r ⃗ ( t ) \vec{r}(t) r ( t ) r ⃗ ( t ) = ∫ 0 t v ⃗ ( t ′ ) d t ′ = ∫ 0 t ( 1 2 t ′ 2 i ^ + t ′ j ^ ) d t ′ = 1 6 t 3 i ^ + 1 2 t 2 j ^ \vec{r}(t) = \int_0^t \vec{v}(t')\,dt' = \int_0^t \left(\frac{1}{2}{t'}^2\,\hat{i} + t'\,\hat{j}\right)dt' = \frac{1}{6}t^3\,\hat{i} + \frac{1}{2}t^2\,\hat{j} r ( t ) = ∫ 0 t v ( t ′ ) d t ′ = ∫ 0 t ( 2 1 t ′ 2 i ^ + t ′ j ^ ) d t ′ = 6 1 t 3 i ^ + 2 1 t 2 j ^
At t = 1.0 s t = 1.0\text{ s} t = 1.0 s r ⃗ ( 1.0 ) = 1 6 i ^ + 1 2 j ^ m \vec{r}(1.0) = \frac{1}{6}\,\hat{i} + \frac{1}{2}\,\hat{j}\text{ m} r ( 1.0 ) = 6 1 i ^ + 2 1 j ^ m
The magnitude of displacement at t = 1.0 s t = 1.0\text{ s} t = 1.0 s ∣ r ⃗ ( 1.0 ) ∣ = ( 1 6 ) 2 + ( 1 2 ) 2 = 1 36 + 1 4 = 10 36 = 10 6 m ≠ 1 6 m |\vec{r}(1.0)| = \sqrt{\left(\frac{1}{6}\right)^2 + \left(\frac{1}{2}\right)^2} = \sqrt{\frac{1}{36} + \frac{1}{4}} = \sqrt{\frac{10}{36}} = \frac{\sqrt{10}}{6}\text{ m} \neq \frac{1}{6}\text{ m} ∣ r ( 1.0 ) ∣ = ( 6 1 ) 2 + ( 2 1 ) 2 = 36 1 + 4 1 = 36 10 = 6 10 m = 6 1 m Option D is incorrect .
2. Torque About the Origin The torque τ ⃗ \vec{\tau} τ τ ⃗ = r ⃗ × F ⃗ \vec{\tau} = \vec{r} \times \vec{F} τ = r × F
At t = 1.0 s t = 1.0\text{ s} t = 1.0 s r ⃗ = 1 6 i ^ + 1 2 j ^ \vec{r} = \frac{1}{6}\,\hat{i} + \frac{1}{2}\,\hat{j} r = 6 1 i ^ + 2 1 j ^ F ⃗ = 1.0 i ^ + 1.0 j ^ = i ^ + j ^ \vec{F} = 1.0\,\hat{i} + 1.0\,\hat{j} = \hat{i} + \hat{j} F = 1.0 i ^ + 1.0 j ^ = i ^ + j ^
Calculating the cross product:
τ ⃗ = ( 1 6 i ^ + 1 2 j ^ ) × ( i ^ + j ^ ) \vec{\tau} = \left(\frac{1}{6}\,\hat{i} + \frac{1}{2}\,\hat{j}\right) \times (\hat{i} + \hat{j}) τ = ( 6 1 i ^ + 2 1 j ^ ) × ( i ^ + j ^ ) τ ⃗ = ( 1 6 ) ( i ^ × j ^ ) + ( 1 2 ) ( j ^ × i ^ ) = 1 6 k ^ − 1 2 k ^ = − 1 3 k ^ N m \vec{\tau} = \left(\frac{1}{6}\right)(\hat{i} \times \hat{j}) + \left(\frac{1}{2}\right)(\hat{j} \times \hat{i}) = \frac{1}{6}\,\hat{k} - \frac{1}{2}\,\hat{k} = -\frac{1}{3}\,\hat{k}\text{ N m} τ = ( 6 1 ) ( i ^ × j ^ ) + ( 2 1 ) ( j ^ × i ^ ) = 6 1 k ^ − 2 1 k ^ = − 3 1 k ^ N m
The magnitude of the torque is:
∣ τ ⃗ ∣ = ∣ − 1 3 k ^ ∣ = 1 3 N m |\vec{\tau}| = \left|-\frac{1}{3}\,\hat{k}\right| = \frac{1}{3}\text{ N m} ∣ τ ∣ = − 3 1 k ^ = 3 1 N m Option A is correct .
The direction of the torque vector τ ⃗ \vec{\tau} τ − k ^ -\hat{k} − k ^ + k ^ +\hat{k} + k ^ Option B is incorrect .
Conclusion The correct statements are A and C .