Physics, 20.08.2019 22:30 maddylosin1032

Abody of mass m and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius r. prove that the body leaves the sphere when 0 = cos(2/3).

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Answer from: twistedhyperboles

$\theta=\cos^{-1}\dfrac{2}{3}$

Explanation:

Given that,

Mass = m

Radius = r

Angle $\theta=\cos\dfrac{2}{3}$

Prove that,

The body leaves the sphere when $\theta=\cos^{-1}\dfrac{2}{3}$

When the particle was contact

$mg\cos\theta=\dfrac{mv^2}{R}$

$Rg\cos\theta=v^2$ .....(I)

Using conservation of energy

$mg(R-R\cos\theta)=\dfrac{1}{2}mv^2$

$v^2=2gR(1-\cos\theta)$ ....(II)

From equation (I) and (II)

$Rg\cos\theta=2gR(1-\cos\theta)$

$\cos\theta=2-2\cos\theta$

$\theta=\cos^{-1}\dfrac{2}{3}$

The body leaves the sphere when $\theta=\cos^{-1}\dfrac{2}{3}$

Hence proved.

Answer from: Quest

a) electrical > light, sound, and thermal< < < <

) the mechanical energy of the firefighter moving.< < < < a) electrical > light, sound, and thermal< < < <

c) both a and b.

Answer from: Quest

force is based upon mass and acceleration.the statement is not always true.

it can be explained from newton's second law of motion.

as per newton's second law of motion the rate of change of momentum is directly proportional to the applied force and takes place along the direction of force.

mathematically it can be written as -

$f\alpha \frac{dp}{dt}$

⇒ $f= k \frac{dp}{dt}$

here k is the proportionality constant whose value is 1.

$f=\frac{dp}{dt}$

$f= \frac{d[mv]}{dt}$

$f= m\frac{dv}{dt} +v\frac{dm}{dt}$

$f=ma+v\frac{dm}{dt}$ [a is the acceleration=dv/dt]

hence we see that force not only depends on mass and acceleration but also on velocity and rate of change of mass. force f will depend on mass and acceleration only when mass is constant.

hence the statement that force is based upon mass and acceleration is false.

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