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MUKUBA UNIVERSITY PHYSICS DEPARTMENT PHY 250: CLASSICAL MECHANICS AND SPECIAL RELATIVITY-2024/2025 TUTORIAL SHEET 4: KINEMATICS AND DYNAMICS OF A RIGID BODY; GRAVITATION 1. (a) Consider a point mass \( m \) with momentum \( \vec{p} \) rotating at a distance \( r \) about an axis. Starting from the definition of the angular momentum, \( \vec{L}=\vec{r} \times \vec{p} \) of a point mass, show that \( \frac{d \bar{t}}{d t}=\vec{\tau} \); where \( \vec{\tau} \) is the torque. (b) The velocity of a particle of mass \( m \) is \( \vec{v}=2 \hat{i}-3 \hat{j}+\hat{k} \) and its position vector \( \vec{r}= \) \( \hat{i}+2 \hat{j}-3 \hat{k} \), find the angular momentum of the particle (c). A particle is located at a position vector \( \vec{r}=(\hat{i}+3 \hat{j}) \mathrm{m} \), and a force acting on it is \( \vec{F}=(3 \hat{i}+2 \hat{j}) N \). What is the torque about (i) the origin and (ii) the point having coordinates \( (0,6) \mathrm{m} \) ? 2. A door 1 m wide and of mass 15 kg , is hinged at one side so that it can rotate without friction about a vertical axis. The door is initially unlatched. A bullet of mass 10 grams travelling at \( 400 \mathrm{~m} / \mathrm{s} \) lodges exactly into the centre of the door in a direction perpendicular to the plane of the door. Find the angular speed of the door just after the bullet embeds itself in the door. Is kinetic energy conserved? 3. A conical pendulum consists of a bob of mass \( m \) in motion in a circular path as shown in figure 3.1. Fig. 3.1 During the motion, the supporting wire of length \( l \) maintains the constant angle \( \theta \) with the vertical. Show that the magnitude of the angular momentum of the bob about the centre of the circle is \[ L=\left(\frac{m^{2} g l^{3} \sin ^{4} \theta}{\cos \theta}\right)^{2} \]

          MUKUBA UNIVERSITY
PHYSICS DEPARTMENT
PHY 250: CLASSICAL MECHANICS AND SPECIAL RELATIVITY-2024/2025
TUTORIAL SHEET 4: KINEMATICS AND DYNAMICS OF A RIGID BODY; GRAVITATION
1. (a) Consider a point mass \( m \) with momentum \( \vec{p} \) rotating at a distance \( r \) about an axis. Starting from the definition of the angular momentum, \( \vec{L}=\vec{r} \times \vec{p} \) of a point mass, show that \( \frac{d \bar{t}}{d t}=\vec{\tau} \); where \( \vec{\tau} \) is the torque.
(b) The velocity of a particle of mass \( m \) is \( \vec{v}=2 \hat{i}-3 \hat{j}+\hat{k} \) and its position vector \( \vec{r}= \) \( \hat{i}+2 \hat{j}-3 \hat{k} \), find the angular momentum of the particle
(c). A particle is located at a position vector \( \vec{r}=(\hat{i}+3 \hat{j}) \mathrm{m} \), and a force acting on it is \( \vec{F}=(3 \hat{i}+2 \hat{j}) N \). What is the torque about (i) the origin and (ii) the point having coordinates \( (0,6) \mathrm{m} \) ?
2. A door 1 m wide and of mass 15 kg , is hinged at one side so that it can rotate without friction about a vertical axis. The door is initially unlatched. A bullet of mass 10 grams travelling at \( 400 \mathrm{~m} / \mathrm{s} \) lodges exactly into the centre of the door in a direction perpendicular to the plane of the door. Find the angular speed of the door just after the bullet embeds itself in the door. Is kinetic energy conserved?
3. A conical pendulum consists of a bob of mass \( m \) in motion in a circular path as shown in figure 3.1.

Fig. 3.1
During the motion, the supporting wire of length \( l \) maintains the constant angle \( \theta \) with the vertical. Show that the magnitude of the angular momentum of the bob about the centre of the circle is
\[
L=\left(\frac{m^{2} g l^{3} \sin ^{4} \theta}{\cos \theta}\right)^{2}
\]

MUKUBA UNIVERSITY
PHYSICS DEPARTMENT
PHY 250: CLASSICAL MECHANICS AND SPECIAL RELATIVITY-2024/2025
TUTORIAL SHEET 4: KINEMATICS AND DYNAMICS OF A RIGID BODY; GRAVITATION
1. (a) Consider a point mass m with momentum p⃗ rotating at a distance r about an axis. Starting from the definition of the angular momentum, L⃗=r⃗×p⃗ of a point mass, show that (d t̅)/(d t)=τ⃗; where τ⃗ is the torque.
(b) The velocity of a particle of mass m is v⃗=2 î-3 ĵ+k̂ and its position vector r⃗= î+2 ĵ-3 k̂, find the angular momentum of the particle
(c). A particle is located at a position vector r⃗=(î+3 ĵ) m, and a force acting on it is F⃗=(3 î+2 ĵ) N. What is the torque about (i) the origin and (ii) the point having coordinates (0,6) m ?
2. A door 1 m wide and of mass 15 kg , is hinged at one side so that it can rotate without friction about a vertical axis. The door is initially unlatched. A bullet of mass 10 grams travelling at 400 m / s lodges exactly into the centre of the door in a direction perpendicular to the plane of the door. Find the angular speed of the door just after the bullet embeds itself in the door. Is kinetic energy conserved?
3. A conical pendulum consists of a bob of mass m in motion in a circular path as shown in figure 3.1.

Fig. 3.1
During the motion, the supporting wire of length l maintains the constant angle θ with the vertical. Show that the magnitude of the angular momentum of the bob about the centre of the circle is

L=((m^2 g l^3sin ^4θ)/(cosθ))^2

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