Two cars of masses m1 and m2 are moving in circles of radii r1 and r2, respectively. Their speeds are such that they make complete circles in the same time t. The ratio of their centripetal acceleration is :

Engineering

Physics

Acceleration

Circular Motion in Vertical Plane

Horizontal Circular Motion

Question

Two cars of masses m₁ and m₂ are moving in circles of radii r₁ and r₂, respectively. Their speeds are such that they make complete circles in the same time t. The ratio of their centripetal acceleration is :

m₁r₁ : m₂r₂

1 : 1

r₁ : r₂

m₁ : m₂

$\frac{a_{1}}{a_{2}} = \frac{r_{1} ω^{2}}{r_{1} ω^{2}} = \frac{r_{1}}{r_{2}}$

This question involves centripetal acceleration in circular motion. Let's analyze it step by step.

Step 1: Understand Centripetal Acceleration
The centripetal acceleration (a_c) for an object moving in a circle of radius r with speed v is given by: $a_{c} = \frac{v^{2}}{r}$

Step 2: Relate Speed to Time Period
The time period (T) is the time taken to complete one full circle. Since both cars complete circles in the same time t, their time periods are equal: T₁ = T₂ = t.

The speed v can be expressed in terms of time period T and circumference 2πr: $v = \frac{2 π r}{T}$

Step 3: Express Centripetal Acceleration in Terms of Time Period
Substitute the expression for v into the centripetal acceleration formula: $a_{c} = \frac{{(\frac{2 π r}{T})}^{2}}{r} = \frac{4 π^{2} r^{2}}{T^{2} r} = \frac{4 π^{2} r}{T^{2}}$

Step 4: Find the Ratio of Accelerations
For car 1: $a_{c 1} = \frac{4 π^{2} r_{1}}{t^{2}}$
For car 2: $a_{c 2} = \frac{4 π^{2} r_{2}}{t^{2}}$
The ratio is: $\frac{a_{c 1}}{a_{c 2}} = \frac{\frac{4 π^{2} r_{1}}{t^{2}}}{\frac{4 π^{2} r_{2}}{t^{2}}} = \frac{r_{1}}{r_{2}}$

Final Answer: The ratio of their centripetal acceleration is r₁ : r₂.

Key Insight: Notice that the centripetal acceleration depends only on radius when time period is constant. The masses m₁ and m₂ do not appear in the final ratio because centripetal acceleration is purely kinematic and independent of mass.

Related Concepts

Circular Motion: When an object moves in a circular path with constant speed, it experiences centripetal acceleration directed toward the center of the circle. This acceleration is necessary to change the direction of velocity continuously.

Time Period in Circular Motion: The time period T is related to angular velocity ω by ω = 2π/T. For uniform circular motion, all kinematic quantities can be expressed in terms of T.

Important Formulas

Centripetal acceleration: $a_{c} = \frac{v^{2}}{r} = ω r^{2} = \frac{4 π^{2} r}{T^{2}}$

Centripetal force: $F_{c} = m a_{c} = \frac{m v^{2}}{r}$ (Note: While acceleration is independent of mass, the centripetal force required does depend on mass)

Detailed Solution

Related Concepts

Important Formulas

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