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Review Problem. A block of mass m 2.00 kg is re- leased from rest h 0.500 m f...

Mar 19, 2024

Review Problem. A block of mass m 2.00 kg is re- leased from rest h 0.500 m from the surface of a table, at the top of a 30.0° incline as shown in Fig- ure P5.62. The frictionless incline is fixed on a table of height H 2.00 m. (a) Determine the acceleration of the block as it slides down the incline. (b) What is the velocity of the block as it leaves the incline? (c) How far from the table will the block hit the floor? (d) How much time has elapsed between when the block is re- leased and when it hits the floor? (e) Does the mass of the block affect any of the above calculations?

Solution

Determine the acceleration of the block as it slides down the incline. Use the component of gravitational acceleration along the incline:

a = g \sin(\theta)

where

g = 9.8 \, m/s^2

and

\theta = 30.0°

. Thus,

a = 9.8 \sin(30.0°) = 4.9 \, m/s^2

Calculate the velocity of the block as it leaves the incline using the equation

v_f^2 = v_i^2 + 2ad

. The initial velocity

v_i = 0

, the acceleration

a = 4.9 \, m/s^2

, and the displacement

d = h / \sin(\theta) = 0.500 \, m / \sin(30.0°) = 1.00 \, m

. Therefore,

v_f = \sqrt{2 \cdot 4.9 \cdot 1.00} = \sqrt{9.8} \approx 3.13 \, m/s

Find how far from the table the block will hit the floor. Use the horizontal velocity component

v_{x} = v_f \cos(\theta)

and the time of flight

t

to find the horizontal distance

x

. First, find

t

using

t = \sqrt{\frac{2H}{g}} = \sqrt{\frac{2 \cdot 2.00}{9.8}} \approx 0.64 \, s

. Then,

x = v_{x}t = 3.13 \cos(30.0°) \cdot 0.64 \approx 1.73 \, m

Calculate the time elapsed from when the block is released to when it hits the floor. The time it takes to slide down the incline is

t_1 = \sqrt{\frac{2d}{a}} = \sqrt{\frac{2 \cdot 1.00}{4.9}} \approx 0.64 \, s

, and the time to fall from the table to the floor is

t_2 = \sqrt{\frac{2H}{g}} \approx 0.64 \, s

. The total time is

t_{total} = t_1 + t_2 \approx 0.64 + 0.64 = 1.28 \, s

Discuss if the mass of the block affects any of the above calculations. The mass of the block does not affect the acceleration down the incline, the velocity as it leaves the incline, the horizontal distance traveled, or the time of flight because all these quantities are independent of mass in the absence of air resistance

Answer

a: 4.9 m/s^2, b: 3.13 m/s, c: 1.73 m, d: 1.28 s, e: No, the mass does not affect the calculations.

Key Concept

Kinematics and dynamics in the context of motion on an incline and projectile motion.

Explanation

The acceleration down the incline is due to the component of gravitational force along the incline, and the horizontal distance and time of flight are determined by projectile motion equations.

Key Concept

Conservation of Energy and Kinematic Equations.

Explanation

The velocity of the block as it leaves the incline is found using kinematic equations, assuming no energy is lost to friction.

Key Concept

Projectile Motion.

Explanation

The horizontal distance the block travels after leaving the incline is calculated using the horizontal component of its velocity and the time of flight.

Key Concept

Independence of Mass in Kinematic Calculations.

Explanation

In the absence of air resistance, the mass of the block does not affect the acceleration, velocity, distance, or time as these quantities are determined by gravitational acceleration and geometry.