Question:medium
A particle of mass \(m\) is moving around the origin with a constant speed \(v\) along a circular path of radius \(R\). When the particle is at \( (0, R) \), its velocity is \( \mathbf{v} = -v \hat{\mathbf{i}} \). The angular momentum of the particle with respect to the origin is :
A particle of mass \(m\) is moving around the origin with a constant speed \(v\) along a circular path of radius \(R\). When the particle is at \( (0, R) \), its velocity is \( \mathbf{v} = -v \hat{\mathbf{i}} \). The angular momentum of the particle with respect to the origin is :
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Angular momentum \( \mathbf{L} = \mathbf{r} \times \mathbf{p} \). Identify the position vector \( \mathbf{r} \) and the linear momentum vector \( \mathbf{p} = m \mathbf{v} \) from the given information. Then compute the cross product using the properties of unit vectors.
Updated On: Nov 26, 2025
- \( mvR \hat{\mathbf{k}} \)
- \( -mvR \hat{\mathbf{k}} \)
- \( mvR \hat{\mathbf{j}} \)
- \( -mvR \hat{\mathbf{j}} \)
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The Correct Option is A
Solution and Explanation
Step 1: Angular Momentum Formula
Angular momentum L of a particle relative to the origin is the cross product of its position vector r and linear momentum vector p = m v: L = r × p = r × (m v).
Step 2: Position and Velocity Vectors
The particle's position is (0, R), making the position vector r = 0 î + R ĵ = R ĵ. The velocity is v = -v î. The linear momentum vector is p = m (-v î) = -m v î.
Step 3: Compute the Cross Product
The cross product is computed as: L = (R ĵ) × (-m v î) = -m v R (ĵ × î).
Step 4: Cross Product of Unit Vectors
Unit vector cross products follow a cyclic order: î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ. Consequently, ĵ × î = - (î × ĵ) = - k̂.
Step 5: Final Expression for Angular Momentum
Substituting this into the expression for L yields: L = -m v R (-k̂) = m v R k̂.
Therefore, the particle's angular momentum with respect to the origin is: L = m v R k̂.
Angular momentum L of a particle relative to the origin is the cross product of its position vector r and linear momentum vector p = m v: L = r × p = r × (m v).
Step 2: Position and Velocity Vectors
The particle's position is (0, R), making the position vector r = 0 î + R ĵ = R ĵ. The velocity is v = -v î. The linear momentum vector is p = m (-v î) = -m v î.
Step 3: Compute the Cross Product
The cross product is computed as: L = (R ĵ) × (-m v î) = -m v R (ĵ × î).
Step 4: Cross Product of Unit Vectors
Unit vector cross products follow a cyclic order: î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ. Consequently, ĵ × î = - (î × ĵ) = - k̂.
Step 5: Final Expression for Angular Momentum
Substituting this into the expression for L yields: L = -m v R (-k̂) = m v R k̂.
Therefore, the particle's angular momentum with respect to the origin is: L = m v R k̂.
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