Question

A 250-kg crate is on a rough ramp, inclined at 30° above the horizontal. The coefficient of kinetic friction between the crate and ramp is 0.22. A horizontal force of 5000 N is applied to the crate, pushing it up the ramp. What is the acceleration of the crate?

Answer 1

Answer:

$8.35 m/s^2$

Explanation:

We are given that

Mass of crate=250 kg

$\theta=30^{\circ}$

Coefficient of kinetic friction between the crate and ramp=0.22

Horizontal force applied on the crate=5000 N

We have to find the acceleration of the crate.

The normal force acting on the crate

$=N=Fsin\theta+mgcos\theta=5000sin 30^{\circ}+250\times 9.8\times cos30^{\circ}=4623.93 N$

The friction force acting against the motion of crate=$0.22\times 4623.93=1017.26 N$

According to newton's second law, the net force accelerating the crate

$ma=Fcos\theta-(F_f+mgsin\theta)$

$a=\frac{5000cos 30^{\circ}-(1017.26+250\times 9.8 sin30^{\circ}}{250}$

$a=8.35 m/s^2$

Hence, the acceleration of the crate=$8.35 m/s^2$

Answer 2

Answer:

The acceleration of crate is $8.35 \;\rm m/s^{2}$.

Explanation:

Given data:

Mass of crate is, $m=250 \;\rm kg$.

Angle of inclination is, $\theta = 30^{\circ}$.

The coefficient of kinetic friction between crate and ramp is, $\mu = 0.22$.

Magnitude of horizontal force is, $F=5000 \;\rm N$.

In an inclined plane, the net force acting on the crate is given as,

$F_{net}=Fcos\theta-(f+mgsin\theta)\\m a=Fcos\theta-(f+mgsin\theta)$

Here, a is the linear acceleration and g is the gravitational acceleration.

f is the frictional force in an inclined plane and its value is,

$f=\mu \times N$

N is the normal reaction acting on crate in an inclined plane and its value is,

$N=Fsin\theta+mgcos\theta\\N=5000 \times sin30+(250 \times 9.8 \times cos30)\\N=5000 \times sin30+(250 \times 9.8 \times cos30)\\N=4621.76 N$

Then frictional force is,

$f=0.22 \times 4621.76=1016.78 \;\rm N$

Then acceleration is,

$m a=Fcos\theta-(f+mgsin\theta)\\250 \times a=5000 \times cos30-(1016.78+250 \times 9.8 \times sin30)\\a=8.35 \;\rm m/s^{2}$

Thus, the acceleration of crate is $8.35 \;\rm m/s^{2}$.

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