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title: "A block of mass \\(m\\) is initially at rest on a frictionless horizontal \\(xy\\)-plane. During the time interval \\(0 \\le t \\le T\\), a horizontal force \\(\\vec{F}(t)\\) is applied to the block with a time-dependent magnitude \\(F(t) = F_{\\text{max}} \\sin\\left(\\dfrac{\\pi t}{T}\\right)\\), where \\(F_{\\text{max}}\\) and \\(T\\) are positive constants. The direction of the force rotates in the plane such that the angle it makes with the \\(+x\\)-axis is given by \\(\\theta(t) = \\dfrac{\\pi t}{2T}\\). Which of the following expressions correctly represents the integral setup for the \\(x\\)-component of the impulse, \\(J_x\\), delivered to the block over the interval \\(0 \\le t \\le T\\)?"
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url: "https://nerd-notes.com/ubq/120771/"
date_modified: "2026-08-23T04:43:11+00:00"
---

# A block of mass \(m\) is initially at rest on a frictionless horizontal \(xy\)-plane. During the time interval \(0 \le t \le T\), a horizontal force \(\vec{F}(t)\) is applied to the block with a time-dependent magnitude \(F(t) = F_{\text{max}} \sin\left(\dfrac{\pi t}{T}\right)\), where \(F_{\text{max}}\) and \(T\) are positive constants. The direction of the force rotates in the plane such that the angle it makes with the \(+x\)-axis is given by \(\theta(t) = \dfrac{\pi t}{2T}\). Which of the following expressions correctly represents the integral setup for the \(x\)-component of the impulse, \(J_x\), delivered to the block over the interval \(0 \le t \le T\)?

A block of mass \(m\) is initially at rest on a frictionless horizontal \(xy\)-plane. During the time interval \(0 \le t \le T\), a horizontal force \(\vec{F}(t)\) is applied to the block with a time-dependent magnitude \(F(t) = F_{\text{max}} \sin\left(\dfrac{\pi t}{T}\right)\), where \(F_{\text{max}}\) and \(T\) are positive constants. The direction of the force rotates in the plane such that the angle it makes with the \(+x\)-axis is given by \(\theta(t) = \dfrac{\pi t}{2T}\). Which of the following expressions correctly represents the integral setup for the \(x\)-component of the impulse, \(J_x\), delivered to the block over the interval \(0 \le t \le T\)?

![A Cartesian coordinate system showing the horizontal xy-plane from an overhead view. The horizontal axis is labeled x extending to the right, and the vertical axis is labeled y extending upward, intersecting at an origin labeled O. A square block labeled m sits at the origin. A single straight arrow representing the force vector originates from the center of the block, pointing upward and to the right into the first quadrant, labeled \vec{F}(t). A curved arc starting from the positive x-axis and ending at the force arrow is labeled \theta(t). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460190-S9IRqR.jpg)

- **A.** \(J_x = F_{\text{max}} \cos\left(\dfrac{\pi}{4}\right) \int_0^T \sin\left(\dfrac{\pi t}{T}\right) dt\)
- **B.** \(J_x = F_{\text{max}} \int_0^T \sin\left(\dfrac{\pi t}{T}\right) \sin\left(\dfrac{\pi t}{2T}\right) dt\)
- **C.** \(J_x = F_{\text{max}} \int_0^T \sin\left(\dfrac{\pi t}{2T}\right) \cos\left(\dfrac{\pi t}{T}\right) dt\)
- **D.** \(J_x = F_{\text{max}} \int_0^T \sin\left(\dfrac{\pi t}{T}\right) \cos\left(\dfrac{\pi t}{2T}\right) dt\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120771/*