Q] Obtain an expression for maximum safety speed with which a vehicle

1.	Q] Obtain an expression for maximum safety speed with which a vehicle can be safely driven along a curved banked roadAndQ] Show that the angle of banking is independent of mass of vehicle.
Answer» Refer to the attachment first for better understanding! (1) Net force acting on the vehicle, $\to\sf{N cos \theta = mg + f sin \theta } \\$ $\to\sf{N cos \theta- f sin \theta = mg ....(1)} \\$ $\to\sf{N sin \theta + f cos \theta =\dfrac{ mv^<klux>2</klux> }{ r} ....(2)} \\ \\$ Dividing (2) and (1) we get, $\to\sf{\dfrac{N sin \theta + f cos \theta }{N cos \theta - f sin \theta } = \dfrac{v^2}{ rg}}\\ \\$ Now LET's divide the numerator and denominator of the L.H.S by N cos θ, $\to\sf{\dfrac{\dfrac{N sin \theta }{N cos \theta } + \dfrac{f cos \theta }{N cos \theta }}{\dfrac{N cos \theta }{N cos \theta } - \dfrac{f sin \theta }{N cos \theta }} = \dfrac{v^2}{ rg}} \\ \\$ $\to\sf{\dfrac{{tan \theta + \dfrac{f}{N} }} {1- \dfrac{f}{N} tan \theta }= \dfrac{v^2}{ rg}} \\ \\$ We know that, $\to\sf{ f = \mu N }\\ \\$ $\to \sf{\mu = f / N } \\ \\$ Hence, $\to\sf{\dfrac{{tan \theta + \mu }} {1-\mu tan \theta }= \dfrac{v^2}{ rg}}\\ \\$ $\to\boxed{\sf{v = \sqrt{rg\bigg(\dfrac{{tan \theta + \mu }} {1- \mu tan \theta } \bigg)}}} \\ \\$ When there is no FRICTION acting between the road and the tire the coefficient of friction (μ) will be zero. $\to\sf{v = \sqrt{rg\bigg(\dfrac{tan \theta +0 } {1-0 }\bigg) }}\\ \\$ Hence the maximum safety speed with which a vehicle can be safely driven along a curved banked road is, $\to\boxed{\sf{v = \sqrt{rg tan \theta }}}$ (2) On rearranging the above equation we get, $\to\sf{\theta = tan^{-1}\bigg(\dfrac{v^2}{rg}\bigg)}$ From the above equation, we can conclude that the angle of BANKING doesn't DEPEND upon the mass of the vehicle.

Q] Obtain an expression for maximum safety speed with which a vehicle can be safely driven along a curved banked roadAndQ] Show that the angle of banking is independent of mass of vehicle.

Answer»

Refer to the attachment first for better understanding!

(1)

Net force acting on the vehicle,

$\to\sf{N cos \theta = mg + f sin \theta } \\$

$\to\sf{N cos \theta- f sin \theta = mg ....(1)} \\$

$\to\sf{N sin \theta + f cos \theta =\dfrac{ mv^<klux>2</klux> }{ r} ....(2)} \\ \\$

Dividing (2) and (1) we get,

$\to\sf{\dfrac{N sin \theta + f cos \theta }{N cos \theta - f sin \theta } = \dfrac{v^2}{ rg}}\\ \\$

Now LET's divide the numerator and denominator of the L.H.S by N cos θ,

$\to\sf{\dfrac{\dfrac{N sin \theta }{N cos \theta } + \dfrac{f cos \theta }{N cos \theta }}{\dfrac{N cos \theta }{N cos \theta } - \dfrac{f sin \theta }{N cos \theta }} = \dfrac{v^2}{ rg}} \\ \\$

$\to\sf{\dfrac{{tan \theta + \dfrac{f}{N} }} {1- \dfrac{f}{N} tan \theta }= \dfrac{v^2}{ rg}} \\ \\$

We know that,

$\to\sf{ f = \mu N }\\ \\$

$\to \sf{\mu = f / N } \\ \\$

Hence,

$\to\sf{\dfrac{{tan \theta + \mu }} {1-\mu tan \theta }= \dfrac{v^2}{ rg}}\\ \\$

$\to\boxed{\sf{v = \sqrt{rg\bigg(\dfrac{{tan \theta + \mu }} {1- \mu tan \theta } \bigg)}}} \\ \\$

When there is no FRICTION acting between the road and the tire the coefficient of friction (μ) will be zero.

$\to\sf{v = \sqrt{rg\bigg(\dfrac{tan \theta +0 } {1-0 }\bigg) }}\\ \\$

Hence the maximum safety speed with which a vehicle can be safely driven along a curved banked road is,

$\to\boxed{\sf{v = \sqrt{rg tan \theta }}}$

(2)

On rearranging the above equation we get,

$\to\sf{\theta = tan^{-1}\bigg(\dfrac{v^2}{rg}\bigg)}$

From the above equation, we can conclude that the angle of BANKING doesn't DEPEND upon the mass of the vehicle.

Q] Obtain an expression for maximum safety speed with which a vehicle can be safely driven along a curved banked roadAndQ] Show that the angle of banking is independent of mass of vehicle.

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Q] Obtain an expression for maximum safety speed with which a vehicle can be safely driven along a curved banked roadAndQ] Show that the angle of banking is independent of mass of vehicle.​

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Q] Obtain an expression for maximum safety speed with which a vehicle can be safely driven along a curved banked roadAndQ] Show that the angle of banking is independent of mass of vehicle.