Rigid Body Motion – Midterm Exam Rich-context Physics Problems
Question 1: Angular Momentum of a Particle (10 points)
A particle has a position vector
$$\mathbf{r} = 2\hat{i} + 3\hat{j} + 4\hat{k}\ \text{m}$$
and a linear momentum
$$\mathbf{p} = 4\hat{i} - 2\hat{j} + 0\hat{k}\ \text{kg·m/s}.$$
What is the angular momentum of the particle about the origin?
Scoring Rubric (10 points)
- Correct formula for angular momentum, $\mathbf{L} = \mathbf{r} \times \mathbf{p}$ (3 pts)
- Correct setup of the cross product (3 pts)
- Accurate vector calculation (3 pts)
- Correct units and final expression (1 pt)
Question 2: Total Angular Momentum of a System (10 points)
Two particles move in circular paths about the origin, with their linear momentum vectors
perpendicular to their respective position vectors.
Particle A is at a distance of 5 m from the origin and has a linear momentum of
60 kg·m/s.
Particle B is at a distance of 8 m from the origin and has a linear momentum of
40 kg·m/s.
What is the total angular momentum of the system about the origin?
Scoring Rubric (10 points)
- Correct expression for angular momentum magnitude, $L = rp$ (3 pts)
- Correct calculation for Particle A (2 pts)
- Correct calculation for Particle B (2 pts)
- Correct summation of angular momenta (2 pts)
- Proper units and final answer (1 pt)
Question 3: Torque Due to an Applied Force (10 points)
A rigid door is hinged at one end. A force of 15 N is applied at a point
0.7 m from the hinge, perpendicular to the plane of the door.
What is the magnitude of the torque acting about the hinge?
Scoring Rubric (10 points)
- Correct torque formula, $\tau = rF$ (3 pts)
- Proper substitution of given values (3 pts)
- Correct numerical calculation (3 pts)
- Correct unit (N·m) and final answer (1 pt)
Question 4: Effect of Increased Force on Torque (10 points)
The same rigid door is acted upon by a force of 45 N applied perpendicular
to the door at a distance of 0.7 m from the hinge.
What torque is produced about the hinge in this case?
Scoring Rubric (10 points)
- Correct identification of torque relationship (3 pts)
- Correct substitution of force and distance (3 pts)
- Accurate calculation (3 pts)
- Correct unit and final answer (1 pt)
Question 5: Angular Displacement under Constant Angular Acceleration (10 points)
An object starts from rest and reaches an angular velocity of
$$\omega = 30\ \text{rad/s}$$
in a time interval of
$$t = 3\ \text{s},$$
while undergoing constant angular acceleration.
What is the angular displacement of the object during this time interval?
Scoring Rubric (10 points)
- Correct use of angular kinematics equations (3 pts)
- Correct determination of angular acceleration (3 pts)
- Correct computation of angular displacement (3 pts)
- Correct units and final answer (1 pt)