An object of mass 4.00 kg is held motionless at the top of a ramp that is inclined at an angle of 35 degrees relative to horizontal. The object is given a gentle shove giving it an initial velocity of 4.00 m/s down the ramp. Exactly 0.443 seconds after being released, the object is 2.00 m away from its release point.
Draw a free body diagram showing all forces on the object.
Determine the acceleration of the object.
Determine the force of friction.
The correct answer and explanation is :
To analyze the motion of the object on the inclined plane, let’s break it down step by step.
Step 1: Free Body Diagram
A free-body diagram (FBD) shows all the forces acting on the object. In this case, the object experiences:
- Gravity (Fg = mg) acting downward.
- Normal Force (Fn) exerted by the surface, perpendicular to the ramp.
- Friction Force (Ff) opposing motion, acting up the ramp.
We’ll generate a free-body diagram image to illustrate these forces.
Step 2: Determine the Acceleration
Using the kinematic equation:
[
s = v_0t + \frac{1}{2} a t^2
]
where:
- ( s = 2.00 ) m (displacement),
- ( v_0 = 4.00 ) m/s (initial velocity),
- ( t = 0.443 ) s (time),
- ( a ) = unknown.
Substituting the values:
[
2.00 = (4.00)(0.443) + \frac{1}{2} a (0.443)^2
]
[
2.00 = 1.772 + \frac{1}{2} a (0.196)
]
[
2.00 – 1.772 = 0.098a
]
[
0.228 = 0.098a
]
[
a = 2.33 \text{ m/s}^2
]
Step 3: Determine the Force of Friction
The forces along the ramp:
[
F_{\text{net}} = m a
]
[
F_{\text{gravity}} = mg \sin \theta = (4.00)(9.81) \sin 35^\circ
]
[
F_{\text{gravity}} = 22.46 \text{ N}
]
Since ( F_{\text{net}} = F_{\text{gravity}} – F_{\text{friction}} ),
[
(4.00)(2.33) = 22.46 – F_{\text{friction}}
]
[
9.32 = 22.46 – F_{\text{friction}}
]
[
F_{\text{friction}} = 13.14 \text{ N}
]
Conclusion
- Acceleration: ( 2.33 ) m/s² down the ramp.
- Friction Force: ( 13.14 ) N opposing motion.
free-body diagram
